Toniolo and Linder, Equation (10+)

Percentage Accurate: 55.2% → 84.1%

Time: 22.2s

Alternatives: 18

Speedup: 2.0×

• 27.6% of points produce a very large (infinite) output. You may want to add a precondition.

Specification

Math

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

FPCore

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

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Initial Program: 55.2% 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: 84.1% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := \frac{t\_m}{\sqrt[3]{\ell}}\\ t_3 := 2 + {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(t\_3 \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{elif}\;t\_m \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(t\_2 \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{t\_2}{\sqrt[3]{\ell}}\right)}^{3}\right) \cdot \left(\tan k \cdot t\_3\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (/ t_m (cbrt l))) (t_3 (+ 2.0 (pow (/ k t_m) 2.0))))
   (*
    t_s
    (if (<= t_m 1.2e-26)
      (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
      (if (<= t_m 1.2e+98)
        (/ 2.0 (/ (* (tan k) (* t_3 (* (sin k) (/ (pow t_m 3.0) l)))) l))
        (if (<= t_m 4e+176)
          (/
           2.0
           (/ (* (tan k) (pow (* t_2 (* (cbrt (sin k)) (cbrt 2.0))) 3.0)) l))
          (/
           2.0
           (* (* (sin k) (pow (/ t_2 (cbrt l)) 3.0)) (* (tan k) 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 / cbrt(l);
	double t_3 = 2.0 + pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.2e-26) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else if (t_m <= 1.2e+98) {
		tmp = 2.0 / ((tan(k) * (t_3 * (sin(k) * (pow(t_m, 3.0) / l)))) / l);
	} else if (t_m <= 4e+176) {
		tmp = 2.0 / ((tan(k) * pow((t_2 * (cbrt(sin(k)) * cbrt(2.0))), 3.0)) / l);
	} else {
		tmp = 2.0 / ((sin(k) * pow((t_2 / cbrt(l)), 3.0)) * (tan(k) * 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 / Math.cbrt(l);
	double t_3 = 2.0 + Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 1.2e-26) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else if (t_m <= 1.2e+98) {
		tmp = 2.0 / ((Math.tan(k) * (t_3 * (Math.sin(k) * (Math.pow(t_m, 3.0) / l)))) / l);
	} else if (t_m <= 4e+176) {
		tmp = 2.0 / ((Math.tan(k) * Math.pow((t_2 * (Math.cbrt(Math.sin(k)) * Math.cbrt(2.0))), 3.0)) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * Math.pow((t_2 / Math.cbrt(l)), 3.0)) * (Math.tan(k) * 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(t_m / cbrt(l))
	t_3 = Float64(2.0 + (Float64(k / t_m) ^ 2.0))
	tmp = 0.0
	if (t_m <= 1.2e-26)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	elseif (t_m <= 1.2e+98)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(t_3 * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))) / l));
	elseif (t_m <= 4e+176)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(t_2 * Float64(cbrt(sin(k)) * cbrt(2.0))) ^ 3.0)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * (Float64(t_2 / cbrt(l)) ^ 3.0)) * Float64(tan(k) * 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[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 1.2e-26], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.2e+98], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(t$95$3 * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 4e+176], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(t$95$2 * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[Power[N[(t$95$2 / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * t$95$3), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]), $MachinePrecision]]]

Derivation

1. Split input into 4 regimes

2. if t < 1.2e-26

   1. Initial program 46.2%

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

      1. associate-/r* 50.5%

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

      2. +-commutative 50.5%

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

      3. associate-+r+ 50.5%

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

      4. metadata-eval 50.5%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-*l/ 51.5%

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

      8. associate-*r/ 51.8%

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

   4. Applied egg-rr 51.8%

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

      1. associate-*l* 51.9%

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

      2. *-commutative 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in k around inf 72.4%

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

   if 1.2e-26 < t < 1.1999999999999999e98

   1. Initial program 89.7%

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

      1. associate-/r* 93.0%

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

      2. +-commutative 93.0%

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

      3. associate-+r+ 93.0%

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

      4. metadata-eval 93.0%

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

      5. associate-*r* 93.0%

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

      6. *-commutative 93.0%

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

      7. associate-*l/ 96.4%

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

      8. associate-*r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l* 99.7%

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

      2. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 1.1999999999999999e98 < t < 4e176

   1. Initial program 53.7%

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

      1. associate-/r* 58.5%

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

      2. +-commutative 58.5%

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

      3. associate-+r+ 58.5%

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

      4. metadata-eval 58.5%

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

      5. associate-*r* 58.5%

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

      6. *-commutative 58.5%

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

      7. associate-*l/ 58.6%

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

      8. associate-*r/ 58.6%

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

   4. Applied egg-rr 58.6%

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

      1. associate-*l* 58.6%

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

      2. *-commutative 58.6%

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

   6. Simplified 58.6%

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

      1. add-cube-cbrt 58.5%

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

      2. pow3 58.5%

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

   8. Applied egg-rr 94.6%

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

   9. Taylor expanded in t around inf 46.9%

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

       1. unpow1/3 94.8%

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

       2. *-lft-identity 94.8%

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

   11. Simplified 94.8%

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

   if 4e176 < t

   1. Initial program 70.9%

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

      1. associate-*l* 70.9%

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

      2. sqr-neg 70.9%

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

      3. sqr-neg 70.9%

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

      4. associate-/r* 72.1%

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

      5. distribute-rgt-in 72.1%

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

      6. unpow2 72.1%

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

      7. times-frac 53.6%

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

      8. sqr-neg 53.6%

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

      9. times-frac 72.1%

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

      10. unpow2 72.1%

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

      11. distribute-rgt-in 72.1%

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

   3. Simplified 72.1%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-div 72.1%

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

      4. rem-cbrt-cube 72.1%

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

   6. Applied egg-rr 72.1%

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

      1. add-cube-cbrt 72.1%

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

      2. pow3 72.1%

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

      3. cbrt-div 72.1%

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

      4. unpow3 72.1%

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

      5. add-cbrt-cube 93.7%

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

   8. Applied egg-rr 93.7%

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

4. Final simplification 78.7%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.2 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.2 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{elif}\;t \leq 4 \cdot 10^{+176}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot {\left(\frac{\frac{t}{\sqrt[3]{\ell}}}{\sqrt[3]{\ell}}\right)}^{3}\right) \cdot \left(\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 2: 78.3% accurate, 0.5× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0))))
   (*
    t_s
    (if (<= (* t_2 (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l))))) INFINITY)
      (/ 2.0 (* t_2 (* (tan k) (* (sin k) (* (/ (pow t_m 2.0) l) (/ t_m l))))))
      (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if ((t_2 * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))))) <= ((double) INFINITY)) {
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * ((pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = 1.0 + (Math.pow((k / t_m), 2.0) + 1.0);
	double tmp;
	if ((t_2 * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))))) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / (t_2 * (Math.tan(k) * (Math.sin(k) * ((Math.pow(t_m, 2.0) / l) * (t_m / l)))));
	} else {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = 1.0 + (math.pow((k / t_m), 2.0) + 1.0)
	tmp = 0
	if (t_2 * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))) <= math.inf:
		tmp = 2.0 / (t_2 * (math.tan(k) * (math.sin(k) * ((math.pow(t_m, 2.0) / l) * (t_m / l)))))
	else:
		tmp = 2.0 / ((math.tan(k) * ((math.pow(k, 2.0) * (t_m * math.sin(k))) / l)) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))
	tmp = 0.0
	if (Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))))) <= Inf)
		tmp = Float64(2.0 / Float64(t_2 * Float64(tan(k) * Float64(sin(k) * Float64(Float64((t_m ^ 2.0) / l) * Float64(t_m / l))))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = 1.0 + (((k / t_m) ^ 2.0) + 1.0);
	tmp = 0.0;
	if ((t_2 * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l))))) <= Inf)
		tmp = 2.0 / (t_2 * (tan(k) * (sin(k) * (((t_m ^ 2.0) / l) * (t_m / l)))));
	else
		tmp = 2.0 / ((tan(k) * (((k ^ 2.0) * (t_m * sin(k))) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 79.6%

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

      1. unpow3 79.6%

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

      2. times-frac 87.4%

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

      3. pow2 87.4%

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

   4. Applied egg-rr 87.4%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. associate-/r* 4.2%

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

      2. +-commutative 4.2%

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

      3. associate-+r+ 4.2%

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

      4. metadata-eval 4.2%

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

      5. associate-*r* 4.2%

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

      6. *-commutative 4.2%

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

      7. associate-*l/ 4.2%

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

      8. associate-*r/ 4.3%

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

   4. Applied egg-rr 4.3%

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

      1. associate-*l* 4.3%

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

      2. *-commutative 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in k around inf 70.5%

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

4. Final simplification 81.8%

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

Alternative 3: 77.9% accurate, 0.5× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<=
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (tan k) (* (sin k) (/ (pow t_m 3.0) (* l l)))))
         INFINITY)
      (/ 2.0 (/ (* (tan k) (* (+ 2.0 t_2) (* (sin k) (/ (pow t_m 3.0) l)))) l))
      (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * (pow(t_m, 3.0) / (l * l))))) <= ((double) INFINITY)) {
		tmp = 2.0 / ((tan(k) * ((2.0 + t_2) * (sin(k) * (pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.sin(k) * (Math.pow(t_m, 3.0) / (l * l))))) <= Double.POSITIVE_INFINITY) {
		tmp = 2.0 / ((Math.tan(k) * ((2.0 + t_2) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	t_2 = math.pow((k / t_m), 2.0)
	tmp = 0
	if ((1.0 + (t_2 + 1.0)) * (math.tan(k) * (math.sin(k) * (math.pow(t_m, 3.0) / (l * l))))) <= math.inf:
		tmp = 2.0 / ((math.tan(k) * ((2.0 + t_2) * (math.sin(k) * (math.pow(t_m, 3.0) / l)))) / l)
	else:
		tmp = 2.0 / ((math.tan(k) * ((math.pow(k, 2.0) * (t_m * math.sin(k))) / l)) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64(sin(k) * Float64((t_m ^ 3.0) / Float64(l * l))))) <= Inf)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(2.0 + t_2) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	t_2 = (k / t_m) ^ 2.0;
	tmp = 0.0;
	if (((1.0 + (t_2 + 1.0)) * (tan(k) * (sin(k) * ((t_m ^ 3.0) / (l * l))))) <= Inf)
		tmp = 2.0 / ((tan(k) * ((2.0 + t_2) * (sin(k) * ((t_m ^ 3.0) / l)))) / l);
	else
		tmp = 2.0 / ((tan(k) * (((k ^ 2.0) * (t_m * sin(k))) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

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

Derivation

1. Split input into 2 regimes

2. if (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) < +inf.0

   1. Initial program 79.6%

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

      1. associate-/r* 83.5%

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

      2. +-commutative 83.5%

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

      3. associate-+r+ 83.5%

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

      4. metadata-eval 83.5%

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

      5. associate-*r* 84.1%

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

      6. *-commutative 84.1%

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

      7. associate-*l/ 85.2%

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

      8. associate-*r/ 86.1%

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

   4. Applied egg-rr 86.1%

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

      1. associate-*l* 86.1%

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

      2. *-commutative 86.1%

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

   6. Simplified 86.1%

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

   if +inf.0 < (*.f64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))

   1. Initial program 0.0%

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

      1. associate-/r* 4.2%

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

      2. +-commutative 4.2%

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

      3. associate-+r+ 4.2%

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

      4. metadata-eval 4.2%

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

      5. associate-*r* 4.2%

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

      6. *-commutative 4.2%

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

      7. associate-*l/ 4.2%

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

      8. associate-*r/ 4.3%

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

   4. Applied egg-rr 4.3%

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

      1. associate-*l* 4.3%

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

      2. *-commutative 4.3%

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

   6. Simplified 4.3%

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

   7. Taylor expanded in k around inf 70.5%

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

4. Final simplification 80.9%

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

Alternative 4: 82.7% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\frac{t\_m}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 6.2e-27)
    (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
    (if (<= t_m 2.7e+98)
      (/
       2.0
       (/
        (*
         (tan k)
         (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (/ (pow t_m 3.0) l))))
        l))
      (/
       2.0
       (/
        (*
         (tan k)
         (pow (* (/ t_m (cbrt l)) (* (cbrt (sin k)) (cbrt 2.0))) 3.0))
        l))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e-27) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else if (t_m <= 2.7e+98) {
		tmp = 2.0 / ((tan(k) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * (pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * pow(((t_m / cbrt(l)) * (cbrt(sin(k)) * cbrt(2.0))), 3.0)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 6.2e-27) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else if (t_m <= 2.7e+98) {
		tmp = 2.0 / ((Math.tan(k) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * Math.pow(((t_m / Math.cbrt(l)) * (Math.cbrt(Math.sin(k)) * Math.cbrt(2.0))), 3.0)) / l);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 6.2e-27)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	elseif (t_m <= 2.7e+98)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(Float64(t_m / cbrt(l)) * Float64(cbrt(sin(k)) * cbrt(2.0))) ^ 3.0)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 6.2e-27], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 2.7e+98], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision] * N[(N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 6.1999999999999997e-27

   1. Initial program 46.2%

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

      1. associate-/r* 50.5%

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

      2. +-commutative 50.5%

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

      3. associate-+r+ 50.5%

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

      4. metadata-eval 50.5%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-*l/ 51.5%

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

      8. associate-*r/ 51.8%

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

   4. Applied egg-rr 51.8%

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

      1. associate-*l* 51.9%

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

      2. *-commutative 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in k around inf 72.4%

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

   if 6.1999999999999997e-27 < t < 2.7e98

   1. Initial program 89.7%

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

      1. associate-/r* 93.0%

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

      2. +-commutative 93.0%

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

      3. associate-+r+ 93.0%

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

      4. metadata-eval 93.0%

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

      5. associate-*r* 93.0%

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

      6. *-commutative 93.0%

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

      7. associate-*l/ 96.4%

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

      8. associate-*r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l* 99.7%

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

      2. *-commutative 99.7%

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

   6. Simplified 99.7%

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

   if 2.7e98 < t

   1. Initial program 61.4%

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

      1. associate-/r* 64.6%

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

      2. +-commutative 64.6%

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

      3. associate-+r+ 64.6%

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

      4. metadata-eval 64.6%

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

      5. associate-*r* 64.6%

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

      6. *-commutative 64.6%

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

      7. associate-*l/ 64.6%

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

      8. associate-*r/ 64.6%

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

   4. Applied egg-rr 64.6%

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

      1. associate-*l* 64.6%

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

      2. *-commutative 64.6%

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

   6. Simplified 64.6%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

   8. Applied egg-rr 87.0%

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

   9. Taylor expanded in t around inf 52.1%

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

       1. unpow1/3 87.1%

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

       2. *-lft-identity 87.1%

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

   11. Simplified 87.1%

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

4. Final simplification 77.6%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{+98}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\frac{t}{\sqrt[3]{\ell}} \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 5: 84.0% accurate, 0.7× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right)} \cdot \frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 8e-26)
    (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
    (/
     2.0
     (/
      (*
       (tan k)
       (pow
        (* (cbrt (* (sin k) (+ 2.0 (pow (/ k t_m) 2.0)))) (/ t_m (cbrt l)))
        3.0))
      l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-26) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * pow((cbrt((sin(k) * (2.0 + pow((k / t_m), 2.0)))) * (t_m / cbrt(l))), 3.0)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 8e-26) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * Math.pow((Math.cbrt((Math.sin(k) * (2.0 + Math.pow((k / t_m), 2.0)))) * (t_m / Math.cbrt(l))), 3.0)) / l);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 8e-26)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(cbrt(Float64(sin(k) * Float64(2.0 + (Float64(k / t_m) ^ 2.0)))) * Float64(t_m / cbrt(l))) ^ 3.0)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 8e-26], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] * N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 8.0000000000000003e-26

   1. Initial program 46.5%

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

      1. associate-/r* 50.7%

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

      2. +-commutative 50.7%

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

      3. associate-+r+ 50.7%

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

      4. metadata-eval 50.7%

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

      5. associate-*r* 51.2%

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

      6. *-commutative 51.2%

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

      7. associate-*l/ 51.7%

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

      8. associate-*r/ 52.1%

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

   4. Applied egg-rr 52.1%

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

      1. associate-*l* 52.1%

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

      2. *-commutative 52.1%

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

   6. Simplified 52.1%

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

   7. Taylor expanded in k around inf 72.6%

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

   if 8.0000000000000003e-26 < t

   1. Initial program 73.0%

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

      1. associate-/r* 76.3%

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

      2. +-commutative 76.3%

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

      3. associate-+r+ 76.3%

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

      4. metadata-eval 76.3%

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

      5. associate-*r* 76.3%

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

      6. *-commutative 76.3%

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

      7. associate-*l/ 77.8%

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

      8. associate-*r/ 79.2%

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

   4. Applied egg-rr 79.2%

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

      1. associate-*l* 79.2%

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

      2. *-commutative 79.2%

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

   6. Simplified 79.2%

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

      1. add-cube-cbrt 79.0%

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

      2. pow3 79.0%

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

   8. Applied egg-rr 92.1%

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

4. Final simplification 77.5%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 8 \cdot 10^{-26}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\sqrt[3]{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)} \cdot \frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 6: 81.2% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t\_m}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t\_m}\right)}^{2} + 1\right)\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.55e-7)
    (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
    (/
     2.0
     (*
      (* (tan k) (* (sin k) (pow (/ (pow t_m 1.5) l) 2.0)))
      (+ 1.0 (+ (pow (/ k t_m) 2.0) 1.0)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.55e-7) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else {
		tmp = 2.0 / ((tan(k) * (sin(k) * pow((pow(t_m, 1.5) / l), 2.0))) * (1.0 + (pow((k / t_m), 2.0) + 1.0)));
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.55d-7) then
        tmp = 2.0d0 / ((tan(k) * (((k ** 2.0d0) * (t_m * sin(k))) / l)) / l)
    else
        tmp = 2.0d0 / ((tan(k) * (sin(k) * (((t_m ** 1.5d0) / l) ** 2.0d0))) * (1.0d0 + (((k / t_m) ** 2.0d0) + 1.0d0)))
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.55e-7) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * (Math.sin(k) * Math.pow((Math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (Math.pow((k / t_m), 2.0) + 1.0)));
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.55e-7:
		tmp = 2.0 / ((math.tan(k) * ((math.pow(k, 2.0) * (t_m * math.sin(k))) / l)) / l)
	else:
		tmp = 2.0 / ((math.tan(k) * (math.sin(k) * math.pow((math.pow(t_m, 1.5) / l), 2.0))) * (1.0 + (math.pow((k / t_m), 2.0) + 1.0)))
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.55e-7)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(sin(k) * (Float64((t_m ^ 1.5) / l) ^ 2.0))) * Float64(1.0 + Float64((Float64(k / t_m) ^ 2.0) + 1.0))));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.55e-7)
		tmp = 2.0 / ((tan(k) * (((k ^ 2.0) * (t_m * sin(k))) / l)) / l);
	else
		tmp = 2.0 / ((tan(k) * (sin(k) * (((t_m ^ 1.5) / l) ^ 2.0))) * (1.0 + (((k / t_m) ^ 2.0) + 1.0)));
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.55e-7], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[Power[N[(N[Power[t$95$m, 1.5], $MachinePrecision] / l), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(1.0 + N[(N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.55e-7

   1. Initial program 47.0%

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

      1. associate-/r* 51.2%

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

      2. +-commutative 51.2%

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

      3. associate-+r+ 51.2%

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

      4. metadata-eval 51.2%

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

      5. associate-*r* 51.7%

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

      6. *-commutative 51.7%

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

      7. associate-*l/ 52.2%

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

      8. associate-*r/ 52.6%

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

   4. Applied egg-rr 52.6%

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

      1. associate-*l* 52.6%

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

      2. *-commutative 52.6%

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

   6. Simplified 52.6%

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

   7. Taylor expanded in k around inf 72.9%

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

   if 1.55e-7 < t

   1. Initial program 72.2%

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

      1. associate-/r* 75.5%

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

      2. add-sqr-sqrt 75.5%

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

      3. pow2 75.5%

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

      4. associate-/r* 72.1%

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

      5. sqrt-div 72.1%

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

      6. sqrt-pow1 81.4%

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

      7. metadata-eval 81.4%

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

      8. sqrt-prod 54.4%

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

      9. add-sqr-sqrt 86.2%

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

   4. Applied egg-rr 86.2%

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

4. Final simplification 76.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.55 \cdot 10^{-7}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\tan k \cdot \left(\sin k \cdot {\left(\frac{{t}^{1.5}}{\ell}\right)}^{2}\right)\right) \cdot \left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 7: 81.4% accurate, 0.8× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t\_m}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\sqrt[3]{\frac{\sin k}{\ell}} \cdot \left(t\_m \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.8e-27)
    (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
    (if (<= t_m 1.9e+94)
      (/
       2.0
       (/
        (*
         (tan k)
         (* (+ 2.0 (pow (/ k t_m) 2.0)) (* (sin k) (/ (pow t_m 3.0) l))))
        l))
      (/
       2.0
       (/
        (* (tan k) (pow (* (cbrt (/ (sin k) l)) (* t_m (cbrt 2.0))) 3.0))
        l))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.8e-27) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else if (t_m <= 1.9e+94) {
		tmp = 2.0 / ((tan(k) * ((2.0 + pow((k / t_m), 2.0)) * (sin(k) * (pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((tan(k) * pow((cbrt((sin(k) / l)) * (t_m * cbrt(2.0))), 3.0)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.8e-27) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else if (t_m <= 1.9e+94) {
		tmp = 2.0 / ((Math.tan(k) * ((2.0 + Math.pow((k / t_m), 2.0)) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((Math.tan(k) * Math.pow((Math.cbrt((Math.sin(k) / l)) * (t_m * Math.cbrt(2.0))), 3.0)) / l);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.8e-27)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	elseif (t_m <= 1.9e+94)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(2.0 + (Float64(k / t_m) ^ 2.0)) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * (Float64(cbrt(Float64(sin(k) / l)) * Float64(t_m * cbrt(2.0))) ^ 3.0)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.8e-27], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 1.9e+94], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[Power[N[(N[Power[N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision], 1/3], $MachinePrecision] * N[(t$95$m * N[Power[2.0, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 5.80000000000000008e-27

   1. Initial program 46.2%

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

      1. associate-/r* 50.5%

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

      2. +-commutative 50.5%

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

      3. associate-+r+ 50.5%

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

      4. metadata-eval 50.5%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-*l/ 51.5%

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

      8. associate-*r/ 51.8%

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

   4. Applied egg-rr 51.8%

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

      1. associate-*l* 51.9%

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

      2. *-commutative 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in k around inf 72.4%

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

   if 5.80000000000000008e-27 < t < 1.8999999999999998e94

   1. Initial program 92.2%

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

      1. associate-/r* 92.2%

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

      2. +-commutative 92.2%

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

      3. associate-+r+ 92.2%

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

      4. metadata-eval 92.2%

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

      5. associate-*r* 92.2%

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

      6. *-commutative 92.2%

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

      7. associate-*l/ 96.0%

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

      8. associate-*r/ 99.8%

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

   4. Applied egg-rr 99.8%

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

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 1.8999999999999998e94 < t

   1. Initial program 61.9%

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

      1. associate-/r* 67.1%

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

      2. +-commutative 67.1%

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

      3. associate-+r+ 67.1%

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

      4. metadata-eval 67.1%

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

      5. associate-*r* 67.1%

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

      6. *-commutative 67.1%

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

      7. associate-*l/ 67.2%

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

      8. associate-*r/ 67.1%

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

   4. Applied egg-rr 67.1%

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

      1. associate-*l* 67.1%

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

      2. *-commutative 67.1%

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

   6. Simplified 67.1%

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

      1. add-cube-cbrt 67.1%

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

      2. pow3 67.1%

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

   8. Applied egg-rr 87.8%

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

   9. Taylor expanded in t around inf 45.6%

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

       1. unpow1/3 85.8%

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

       2. *-lft-identity 85.8%

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

       3. *-commutative 85.8%

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

   11. Simplified 85.8%

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

4. Final simplification 77.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.8 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+94}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot {\left(\sqrt[3]{\frac{\sin k}{\ell}} \cdot \left(t \cdot \sqrt[3]{2}\right)\right)}^{3}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 8: 80.2% accurate, 1.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ \begin{array}{l} t_2 := {\left(\frac{k}{t\_m}\right)}^{2}\\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t\_m \leq 5.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + t\_2\right) \cdot \left(\sin k \cdot \frac{{t\_m}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left(t\_2 + 1\right)\right) \cdot \left(\tan k \cdot \frac{{\left(t\_m \cdot \sqrt[3]{\frac{k}{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (let* ((t_2 (pow (/ k t_m) 2.0)))
   (*
    t_s
    (if (<= t_m 6.2e-27)
      (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l))
      (if (<= t_m 5.2e+96)
        (/
         2.0
         (/ (* (tan k) (* (+ 2.0 t_2) (* (sin k) (/ (pow t_m 3.0) l)))) l))
        (/
         2.0
         (*
          (+ 1.0 (+ t_2 1.0))
          (* (tan k) (/ (pow (* t_m (cbrt (/ k l))) 3.0) l)))))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double t_2 = pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.2e-27) {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	} else if (t_m <= 5.2e+96) {
		tmp = 2.0 / ((tan(k) * ((2.0 + t_2) * (sin(k) * (pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (tan(k) * (pow((t_m * cbrt((k / l))), 3.0) / l)));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double t_2 = Math.pow((k / t_m), 2.0);
	double tmp;
	if (t_m <= 6.2e-27) {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	} else if (t_m <= 5.2e+96) {
		tmp = 2.0 / ((Math.tan(k) * ((2.0 + t_2) * (Math.sin(k) * (Math.pow(t_m, 3.0) / l)))) / l);
	} else {
		tmp = 2.0 / ((1.0 + (t_2 + 1.0)) * (Math.tan(k) * (Math.pow((t_m * Math.cbrt((k / l))), 3.0) / l)));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	t_2 = Float64(k / t_m) ^ 2.0
	tmp = 0.0
	if (t_m <= 6.2e-27)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	elseif (t_m <= 5.2e+96)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64(2.0 + t_2) * Float64(sin(k) * Float64((t_m ^ 3.0) / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(1.0 + Float64(t_2 + 1.0)) * Float64(tan(k) * Float64((Float64(t_m * cbrt(Float64(k / l))) ^ 3.0) / l))));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := Block[{t$95$2 = N[Power[N[(k / t$95$m), $MachinePrecision], 2.0], $MachinePrecision]}, N[(t$95$s * If[LessEqual[t$95$m, 6.2e-27], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.2e+96], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(2.0 + t$95$2), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[Power[t$95$m, 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(1.0 + N[(t$95$2 + 1.0), $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(N[Power[N[(t$95$m * N[Power[N[(k / l), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]]

Derivation

1. Split input into 3 regimes

2. if t < 6.1999999999999997e-27

   1. Initial program 46.2%

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

      1. associate-/r* 50.5%

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

      2. +-commutative 50.5%

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

      3. associate-+r+ 50.5%

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

      4. metadata-eval 50.5%

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

      5. associate-*r* 50.9%

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

      6. *-commutative 50.9%

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

      7. associate-*l/ 51.5%

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

      8. associate-*r/ 51.8%

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

   4. Applied egg-rr 51.8%

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

      1. associate-*l* 51.9%

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

      2. *-commutative 51.9%

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

   6. Simplified 51.9%

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

   7. Taylor expanded in k around inf 72.4%

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

   if 6.1999999999999997e-27 < t < 5.2e96

   1. Initial program 89.3%

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

      1. associate-/r* 92.7%

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

      2. +-commutative 92.7%

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

      3. associate-+r+ 92.7%

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

      4. metadata-eval 92.7%

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

      5. associate-*r* 92.7%

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

      6. *-commutative 92.7%

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

      7. associate-*l/ 96.3%

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

      8. associate-*r/ 99.7%

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

   4. Applied egg-rr 99.7%

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

      1. associate-*l* 99.8%

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

      2. *-commutative 99.8%

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

   6. Simplified 99.8%

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

   if 5.2e96 < t

   1. Initial program 62.4%

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

      1. associate-/r* 65.5%

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

      2. associate-*l/ 65.5%

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

   4. Applied egg-rr 65.5%

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

   5. Taylor expanded in k around 0 65.4%

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

      1. associate-/l* 64.7%

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

   7. Simplified 64.7%

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

      1. add-cube-cbrt 64.6%

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

      2. pow3 64.6%

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

      3. associate-/r/ 65.2%

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

      4. cbrt-prod 65.2%

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

      5. unpow3 65.3%

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

      6. add-cbrt-cube 77.9%

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

   9. Applied egg-rr 77.9%

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

4. Final simplification 76.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 6.2 \cdot 10^{-27}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \mathbf{elif}\;t \leq 5.2 \cdot 10^{+96}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \frac{{t}^{3}}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(1 + \left({\left(\frac{k}{t}\right)}^{2} + 1\right)\right) \cdot \left(\tan k \cdot \frac{{\left(t \cdot \sqrt[3]{\frac{k}{\ell}}\right)}^{3}}{\ell}\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 9: 66.2% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.05 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\frac{{\ell}^{2}}{t\_m}}}\\ \mathbf{elif}\;t\_m \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t\_m}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.05e-93)
    (/ 2.0 (/ (pow (* k (sin k)) 2.0) (/ (pow l 2.0) t_m)))
    (if (<= t_m 5.6e+102)
      (/ 2.0 (/ (* (tan k) (* 2.0 (* (pow t_m 3.0) (/ k l)))) l))
      (/ 2.0 (* (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l)) (* 2.0 k)))))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-93) {
		tmp = 2.0 / (pow((k * sin(k)), 2.0) / (pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((tan(k) * (2.0 * (pow(t_m, 3.0) * (k / l)))) / l);
	} else {
		tmp = 2.0 / ((sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)) * (2.0 * k));
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.05e-93) {
		tmp = 2.0 / (Math.pow((k * Math.sin(k)), 2.0) / (Math.pow(l, 2.0) / t_m));
	} else if (t_m <= 5.6e+102) {
		tmp = 2.0 / ((Math.tan(k) * (2.0 * (Math.pow(t_m, 3.0) * (k / l)))) / l);
	} else {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)) * (2.0 * k));
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.05e-93)
		tmp = Float64(2.0 / Float64((Float64(k * sin(k)) ^ 2.0) / Float64((l ^ 2.0) / t_m)));
	elseif (t_m <= 5.6e+102)
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 * Float64((t_m ^ 3.0) * Float64(k / l)))) / l));
	else
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)) * Float64(2.0 * k)));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.05e-93], N[(2.0 / N[(N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[t$95$m, 5.6e+102], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]), $MachinePrecision]

Derivation

1. Split input into 3 regimes

2. if t < 1.05e-93

   1. Initial program 43.4%

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

   3. Taylor expanded in t around 0 65.6%

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

      1. times-frac 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

      1. expm1-log1p-u 48.3%

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

      2. expm1-udef 22.7%

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

      3. frac-times 24.6%

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

      4. pow-prod-down 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. expm1-def 47.3%

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

      2. expm1-log1p 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in k around 0 53.7%

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

   if 1.05e-93 < t < 5.60000000000000037e102

   1. Initial program 81.8%

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

      1. associate-/r* 87.8%

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

      2. +-commutative 87.8%

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

      3. associate-+r+ 87.8%

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

      4. metadata-eval 87.8%

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

      5. associate-*r* 89.8%

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

      6. *-commutative 89.8%

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

      7. associate-*l/ 91.9%

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

      8. associate-*r/ 93.8%

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

   4. Applied egg-rr 93.8%

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

      1. associate-*l* 93.9%

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

      2. *-commutative 93.9%

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

   6. Simplified 93.9%

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

   7. Taylor expanded in k around 0 77.4%

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

      1. associate-*l/ 81.6%

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

      2. *-commutative 81.6%

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

   9. Simplified 81.6%

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

   if 5.60000000000000037e102 < t

   1. Initial program 63.5%

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

      1. associate-*l* 63.5%

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

      2. sqr-neg 63.5%

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

      3. sqr-neg 63.5%

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

      4. associate-/r* 64.2%

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

      5. distribute-rgt-in 64.2%

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

      6. unpow2 64.2%

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

      7. times-frac 52.4%

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

      8. sqr-neg 52.4%

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

      9. times-frac 64.2%

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

      10. unpow2 64.2%

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

      11. distribute-rgt-in 64.2%

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

   3. Simplified 64.2%

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

      1. add-cube-cbrt 64.2%

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

      2. pow3 64.2%

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

      3. cbrt-div 64.2%

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

      4. rem-cbrt-cube 80.8%

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

   6. Applied egg-rr 80.8%

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

   7. Taylor expanded in k around 0 75.5%

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

      1. *-commutative 75.5%

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

   9. Simplified 75.5%

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

4. Final simplification 61.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.05 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{{\left(k \cdot \sin k\right)}^{2}}{\frac{{\ell}^{2}}{t}}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{+102}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \end{array} \]
5. Add Preprocessing

Alternative 10: 68.0% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{\left(\frac{t\_m}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t\_m \cdot \sin k\right)}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= k 9.5e-6)
    (/ 2.0 (* (* (sin k) (/ (pow (/ t_m (cbrt l)) 3.0) l)) (* 2.0 k)))
    (/ 2.0 (/ (* (tan k) (/ (* (pow k 2.0) (* t_m (sin k))) l)) l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e-6) {
		tmp = 2.0 / ((sin(k) * (pow((t_m / cbrt(l)), 3.0) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((tan(k) * ((pow(k, 2.0) * (t_m * sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (k <= 9.5e-6) {
		tmp = 2.0 / ((Math.sin(k) * (Math.pow((t_m / Math.cbrt(l)), 3.0) / l)) * (2.0 * k));
	} else {
		tmp = 2.0 / ((Math.tan(k) * ((Math.pow(k, 2.0) * (t_m * Math.sin(k))) / l)) / l);
	}
	return t_s * tmp;
}

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (k <= 9.5e-6)
		tmp = Float64(2.0 / Float64(Float64(sin(k) * Float64((Float64(t_m / cbrt(l)) ^ 3.0) / l)) * Float64(2.0 * k)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(Float64((k ^ 2.0) * Float64(t_m * sin(k))) / l)) / l));
	end
	return Float64(t_s * tmp)
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[k, 9.5e-6], N[(2.0 / N[(N[(N[Sin[k], $MachinePrecision] * N[(N[Power[N[(t$95$m / N[Power[l, 1/3], $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] * N[(2.0 * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[(t$95$m * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if k < 9.5000000000000005e-6

   1. Initial program 56.9%

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

      1. associate-*l* 57.3%

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

      2. sqr-neg 57.3%

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

      3. sqr-neg 57.3%

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

      4. associate-/r* 62.5%

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

      5. distribute-rgt-in 62.5%

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

      6. unpow2 62.5%

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

      7. times-frac 48.6%

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

      8. sqr-neg 48.6%

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

      9. times-frac 62.5%

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

      10. unpow2 62.5%

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

      11. distribute-rgt-in 62.5%

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

   3. Simplified 62.5%

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

      1. add-cube-cbrt 62.4%

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

      2. pow3 62.4%

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

      3. cbrt-div 62.4%

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

      4. rem-cbrt-cube 69.7%

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

   6. Applied egg-rr 69.7%

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

   7. Taylor expanded in k around 0 63.9%

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

      1. *-commutative 63.9%

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

   9. Simplified 63.9%

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

   if 9.5000000000000005e-6 < k

   1. Initial program 41.3%

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

      1. associate-/r* 41.4%

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

      2. +-commutative 41.4%

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

      3. associate-+r+ 41.4%

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

      4. metadata-eval 41.4%

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

      5. associate-*r* 41.4%

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

      6. *-commutative 41.4%

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

      7. associate-*l/ 41.4%

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

      8. associate-*r/ 42.9%

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

   4. Applied egg-rr 42.9%

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

      1. associate-*l* 42.9%

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

      2. *-commutative 42.9%

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

   6. Simplified 42.9%

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

   7. Taylor expanded in k around inf 86.2%

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

4. Final simplification 69.2%

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq 9.5 \cdot 10^{-6}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \frac{{\left(\frac{t}{\sqrt[3]{\ell}}\right)}^{3}}{\ell}\right) \cdot \left(2 \cdot k\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \frac{{k}^{2} \cdot \left(t \cdot \sin k\right)}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 11: 63.8% accurate, 1.3× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\ell}^{2} \cdot \frac{\cos k}{t\_m}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t\_m}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.1e-93)
    (/ 2.0 (/ (pow k 4.0) (* (pow l 2.0) (/ (cos k) t_m))))
    (/ 2.0 (/ (* (tan k) (* 2.0 (* (pow t_m 3.0) (/ k l)))) l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-93) {
		tmp = 2.0 / (pow(k, 4.0) / (pow(l, 2.0) * (cos(k) / t_m)));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 * (pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.1d-93) then
        tmp = 2.0d0 / ((k ** 4.0d0) / ((l ** 2.0d0) * (cos(k) / t_m)))
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 * ((t_m ** 3.0d0) * (k / l)))) / l)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.1e-93) {
		tmp = 2.0 / (Math.pow(k, 4.0) / (Math.pow(l, 2.0) * (Math.cos(k) / t_m)));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 * (Math.pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.1e-93:
		tmp = 2.0 / (math.pow(k, 4.0) / (math.pow(l, 2.0) * (math.cos(k) / t_m)))
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 * (math.pow(t_m, 3.0) * (k / l)))) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.1e-93)
		tmp = Float64(2.0 / Float64((k ^ 4.0) / Float64((l ^ 2.0) * Float64(cos(k) / t_m))));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 * Float64((t_m ^ 3.0) * Float64(k / l)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.1e-93)
		tmp = 2.0 / ((k ^ 4.0) / ((l ^ 2.0) * (cos(k) / t_m)));
	else
		tmp = 2.0 / ((tan(k) * (2.0 * ((t_m ^ 3.0) * (k / l)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.1e-93], N[(2.0 / N[(N[Power[k, 4.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.09999999999999998e-93

   1. Initial program 43.4%

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

   3. Taylor expanded in t around 0 65.6%

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

      1. times-frac 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

      1. expm1-log1p-u 48.3%

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

      2. expm1-udef 22.7%

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

      3. frac-times 24.6%

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

      4. pow-prod-down 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. expm1-def 47.3%

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

      2. expm1-log1p 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in k around 0 53.4%

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

   if 1.09999999999999998e-93 < t

   1. Initial program 74.0%

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

      1. associate-/r* 77.7%

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

      2. +-commutative 77.7%

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

      3. associate-+r+ 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*r* 78.9%

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

      6. *-commutative 78.9%

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

      7. associate-*l/ 80.1%

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

      8. associate-*r/ 81.2%

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

   4. Applied egg-rr 81.2%

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

      1. associate-*l* 81.2%

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

      2. *-commutative 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 71.8%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   9. Simplified 74.1%

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

4. Final simplification 60.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.1 \cdot 10^{-93}:\\ \;\;\;\;\frac{2}{\frac{{k}^{4}}{{\ell}^{2} \cdot \frac{\cos k}{t}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 12: 63.7% accurate, 1.3× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-94)
    (/ 2.0 (/ (pow (* k (sin k)) 2.0) (/ (pow l 2.0) t_m)))
    (/ 2.0 (/ (* (tan k) (* 2.0 (* (pow t_m 3.0) (/ k l)))) l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-94) {
		tmp = 2.0 / (pow((k * sin(k)), 2.0) / (pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 * (pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-94) then
        tmp = 2.0d0 / (((k * sin(k)) ** 2.0d0) / ((l ** 2.0d0) / t_m))
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 * ((t_m ** 3.0d0) * (k / l)))) / l)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-94) {
		tmp = 2.0 / (Math.pow((k * Math.sin(k)), 2.0) / (Math.pow(l, 2.0) / t_m));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 * (Math.pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-94:
		tmp = 2.0 / (math.pow((k * math.sin(k)), 2.0) / (math.pow(l, 2.0) / t_m))
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 * (math.pow(t_m, 3.0) * (k / l)))) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-94)
		tmp = Float64(2.0 / Float64((Float64(k * sin(k)) ^ 2.0) / Float64((l ^ 2.0) / t_m)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 * Float64((t_m ^ 3.0) * Float64(k / l)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-94)
		tmp = 2.0 / (((k * sin(k)) ^ 2.0) / ((l ^ 2.0) / t_m));
	else
		tmp = 2.0 / ((tan(k) * (2.0 * ((t_m ^ 3.0) * (k / l)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-94], N[(2.0 / N[(N[Power[N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision], 2.0], $MachinePrecision] / N[(N[Power[l, 2.0], $MachinePrecision] / t$95$m), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.9999999999999999e-94

   1. Initial program 43.4%

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

   3. Taylor expanded in t around 0 65.6%

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

      1. times-frac 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

      1. expm1-log1p-u 48.3%

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

      2. expm1-udef 22.7%

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

      3. frac-times 24.6%

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

      4. pow-prod-down 24.6%

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

   7. Applied egg-rr 24.6%

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

      1. expm1-def 47.3%

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

      2. expm1-log1p 64.4%

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

   9. Simplified 64.4%

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

   10. Taylor expanded in k around 0 53.7%

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

   if 1.9999999999999999e-94 < t

   1. Initial program 74.0%

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

      1. associate-/r* 77.7%

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

      2. +-commutative 77.7%

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

      3. associate-+r+ 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*r* 78.9%

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

      6. *-commutative 78.9%

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

      7. associate-*l/ 80.1%

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

      8. associate-*r/ 81.2%

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

   4. Applied egg-rr 81.2%

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

      1. associate-*l* 81.2%

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

      2. *-commutative 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 71.8%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   9. Simplified 74.1%

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

4. Final simplification 60.2%

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

Alternative 13: 63.0% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t\_m}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 2e-95)
    (/ (/ 2.0 t_m) (/ (pow k 4.0) (pow l 2.0)))
    (/ 2.0 (/ (* (tan k) (* 2.0 (* (pow t_m 3.0) (/ k l)))) l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-95) {
		tmp = (2.0 / t_m) / (pow(k, 4.0) / pow(l, 2.0));
	} else {
		tmp = 2.0 / ((tan(k) * (2.0 * (pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 2d-95) then
        tmp = (2.0d0 / t_m) / ((k ** 4.0d0) / (l ** 2.0d0))
    else
        tmp = 2.0d0 / ((tan(k) * (2.0d0 * ((t_m ** 3.0d0) * (k / l)))) / l)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 2e-95) {
		tmp = (2.0 / t_m) / (Math.pow(k, 4.0) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / ((Math.tan(k) * (2.0 * (Math.pow(t_m, 3.0) * (k / l)))) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 2e-95:
		tmp = (2.0 / t_m) / (math.pow(k, 4.0) / math.pow(l, 2.0))
	else:
		tmp = 2.0 / ((math.tan(k) * (2.0 * (math.pow(t_m, 3.0) * (k / l)))) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 2e-95)
		tmp = Float64(Float64(2.0 / t_m) / Float64((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(tan(k) * Float64(2.0 * Float64((t_m ^ 3.0) * Float64(k / l)))) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 2e-95)
		tmp = (2.0 / t_m) / ((k ^ 4.0) / (l ^ 2.0));
	else
		tmp = 2.0 / ((tan(k) * (2.0 * ((t_m ^ 3.0) * (k / l)))) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 2e-95], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[Tan[k], $MachinePrecision] * N[(2.0 * N[(N[Power[t$95$m, 3.0], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.99999999999999998e-95

   1. Initial program 43.4%

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

   3. Taylor expanded in t around 0 65.6%

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

      1. times-frac 64.7%

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

      2. *-commutative 64.7%

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

      3. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in k around 0 50.6%

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

      1. associate-*r/ 50.6%

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

      2. *-commutative 50.6%

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

      3. times-frac 52.3%

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

   8. Simplified 52.3%

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

   9. Taylor expanded in t around 0 50.6%

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

       1. associate-*r/ 50.6%

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

       2. *-commutative 50.6%

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

       3. associate-/r* 52.7%

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

       4. associate-*l/ 52.7%

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

       5. associate-/l* 52.3%

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

   11. Simplified 52.3%

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

   if 1.99999999999999998e-95 < t

   1. Initial program 74.0%

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

      1. associate-/r* 77.7%

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

      2. +-commutative 77.7%

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

      3. associate-+r+ 77.7%

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

      4. metadata-eval 77.7%

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

      5. associate-*r* 78.9%

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

      6. *-commutative 78.9%

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

      7. associate-*l/ 80.1%

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

      8. associate-*r/ 81.2%

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

   4. Applied egg-rr 81.2%

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

      1. associate-*l* 81.2%

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

      2. *-commutative 81.2%

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

   6. Simplified 81.2%

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

   7. Taylor expanded in k around 0 71.8%

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

      1. associate-*l/ 74.1%

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

      2. *-commutative 74.1%

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

   9. Simplified 74.1%

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

4. Final simplification 59.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 2 \cdot 10^{-95}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\tan k \cdot \left(2 \cdot \left({t}^{3} \cdot \frac{k}{\ell}\right)\right)}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 14: 58.7% accurate, 1.9× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 5.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t\_m}^{3}}{\ell}}{\ell}}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 5.3e-104)
    (/ (/ 2.0 t_m) (/ (pow k 4.0) (pow l 2.0)))
    (/ 2.0 (/ (* 2.0 (/ (* (pow k 2.0) (pow t_m 3.0)) l)) l)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.3e-104) {
		tmp = (2.0 / t_m) / (pow(k, 4.0) / pow(l, 2.0));
	} else {
		tmp = 2.0 / ((2.0 * ((pow(k, 2.0) * pow(t_m, 3.0)) / l)) / l);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 5.3d-104) then
        tmp = (2.0d0 / t_m) / ((k ** 4.0d0) / (l ** 2.0d0))
    else
        tmp = 2.0d0 / ((2.0d0 * (((k ** 2.0d0) * (t_m ** 3.0d0)) / l)) / l)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 5.3e-104) {
		tmp = (2.0 / t_m) / (Math.pow(k, 4.0) / Math.pow(l, 2.0));
	} else {
		tmp = 2.0 / ((2.0 * ((Math.pow(k, 2.0) * Math.pow(t_m, 3.0)) / l)) / l);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 5.3e-104:
		tmp = (2.0 / t_m) / (math.pow(k, 4.0) / math.pow(l, 2.0))
	else:
		tmp = 2.0 / ((2.0 * ((math.pow(k, 2.0) * math.pow(t_m, 3.0)) / l)) / l)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 5.3e-104)
		tmp = Float64(Float64(2.0 / t_m) / Float64((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = Float64(2.0 / Float64(Float64(2.0 * Float64(Float64((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 5.3e-104)
		tmp = (2.0 / t_m) / ((k ^ 4.0) / (l ^ 2.0));
	else
		tmp = 2.0 / ((2.0 * (((k ^ 2.0) * (t_m ^ 3.0)) / l)) / l);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 5.3e-104], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(2.0 * N[(N[(N[Power[k, 2.0], $MachinePrecision] * N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 5.30000000000000018e-104

   1. Initial program 42.8%

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

   3. Taylor expanded in t around 0 65.2%

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

      1. times-frac 64.3%

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

      2. *-commutative 64.3%

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

      3. associate-/l* 64.3%

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

   5. Simplified 64.3%

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

   6. Taylor expanded in k around 0 50.1%

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

      1. associate-*r/ 50.1%

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

      2. *-commutative 50.1%

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

      3. times-frac 51.7%

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

   8. Simplified 51.7%

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

   9. Taylor expanded in t around 0 50.1%

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

       1. associate-*r/ 50.1%

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

       2. *-commutative 50.1%

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

       3. associate-/r* 52.2%

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

       4. associate-*l/ 52.2%

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

       5. associate-/l* 51.8%

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

   11. Simplified 51.8%

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

   if 5.30000000000000018e-104 < t

   1. Initial program 74.6%

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

      1. associate-/r* 78.3%

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

      2. +-commutative 78.3%

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

      3. associate-+r+ 78.3%

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

      4. metadata-eval 78.3%

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

      5. associate-*r* 79.4%

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

      6. *-commutative 79.4%

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

      7. associate-*l/ 80.6%

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

      8. associate-*r/ 81.6%

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

   4. Applied egg-rr 81.6%

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

      1. associate-*l* 81.7%

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

      2. *-commutative 81.7%

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

   6. Simplified 81.7%

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

      1. add-cube-cbrt 81.4%

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

      2. pow3 81.4%

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

   8. Applied egg-rr 91.5%

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

   9. Taylor expanded in k around 0 67.6%

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

4. Final simplification 57.0%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 5.3 \cdot 10^{-104}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{2 \cdot \frac{{k}^{2} \cdot {t}^{3}}{\ell}}{\ell}}\\ \end{array} \]
5. Add Preprocessing

Alternative 15: 56.3% accurate, 2.0× speedup

Math

\[\begin{array}{l} t_m = \left|t\right| \\ t_s = \mathsf{copysign}\left(1, t\right) \\ t\_s \cdot \begin{array}{l} \mathbf{if}\;t\_m \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{t\_m}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t\_m}^{3}}}{k \cdot k}\\ \end{array} \end{array} \]

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (*
  t_s
  (if (<= t_m 1.4e-78)
    (/ (/ 2.0 t_m) (/ (pow k 4.0) (pow l 2.0)))
    (/ (/ (pow l 2.0) (pow t_m 3.0)) (* k k)))))

C

t_m = fabs(t);
t_s = copysign(1.0, t);
double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.4e-78) {
		tmp = (2.0 / t_m) / (pow(k, 4.0) / pow(l, 2.0));
	} else {
		tmp = (pow(l, 2.0) / pow(t_m, 3.0)) / (k * k);
	}
	return t_s * tmp;
}

Fortran

t_m = abs(t)
t_s = copysign(1.0d0, t)
real(8) function code(t_s, t_m, l, k)
    real(8), intent (in) :: t_s
    real(8), intent (in) :: t_m
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if (t_m <= 1.4d-78) then
        tmp = (2.0d0 / t_m) / ((k ** 4.0d0) / (l ** 2.0d0))
    else
        tmp = ((l ** 2.0d0) / (t_m ** 3.0d0)) / (k * k)
    end if
    code = t_s * tmp
end function

Java

t_m = Math.abs(t);
t_s = Math.copySign(1.0, t);
public static double code(double t_s, double t_m, double l, double k) {
	double tmp;
	if (t_m <= 1.4e-78) {
		tmp = (2.0 / t_m) / (Math.pow(k, 4.0) / Math.pow(l, 2.0));
	} else {
		tmp = (Math.pow(l, 2.0) / Math.pow(t_m, 3.0)) / (k * k);
	}
	return t_s * tmp;
}

Python

t_m = math.fabs(t)
t_s = math.copysign(1.0, t)
def code(t_s, t_m, l, k):
	tmp = 0
	if t_m <= 1.4e-78:
		tmp = (2.0 / t_m) / (math.pow(k, 4.0) / math.pow(l, 2.0))
	else:
		tmp = (math.pow(l, 2.0) / math.pow(t_m, 3.0)) / (k * k)
	return t_s * tmp

Julia

t_m = abs(t)
t_s = copysign(1.0, t)
function code(t_s, t_m, l, k)
	tmp = 0.0
	if (t_m <= 1.4e-78)
		tmp = Float64(Float64(2.0 / t_m) / Float64((k ^ 4.0) / (l ^ 2.0)));
	else
		tmp = Float64(Float64((l ^ 2.0) / (t_m ^ 3.0)) / Float64(k * k));
	end
	return Float64(t_s * tmp)
end

MATLAB

t_m = abs(t);
t_s = sign(t) * abs(1.0);
function tmp_2 = code(t_s, t_m, l, k)
	tmp = 0.0;
	if (t_m <= 1.4e-78)
		tmp = (2.0 / t_m) / ((k ^ 4.0) / (l ^ 2.0));
	else
		tmp = ((l ^ 2.0) / (t_m ^ 3.0)) / (k * k);
	end
	tmp_2 = t_s * tmp;
end

Wolfram

t_m = N[Abs[t], $MachinePrecision]
t_s = N[With[{TMP1 = Abs[1.0], TMP2 = Sign[t]}, TMP1 * If[TMP2 == 0, 1, TMP2]], $MachinePrecision]
code[t$95$s_, t$95$m_, l_, k_] := N[(t$95$s * If[LessEqual[t$95$m, 1.4e-78], N[(N[(2.0 / t$95$m), $MachinePrecision] / N[(N[Power[k, 4.0], $MachinePrecision] / N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[Power[l, 2.0], $MachinePrecision] / N[Power[t$95$m, 3.0], $MachinePrecision]), $MachinePrecision] / N[(k * k), $MachinePrecision]), $MachinePrecision]]), $MachinePrecision]

Derivation

1. Split input into 2 regimes

2. if t < 1.40000000000000012e-78

   1. Initial program 43.3%

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

   3. Taylor expanded in t around 0 65.6%

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

      1. times-frac 64.8%

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

      2. *-commutative 64.8%

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

      3. associate-/l* 64.8%

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

   5. Simplified 64.8%

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

   6. Taylor expanded in k around 0 50.9%

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

      1. associate-*r/ 50.9%

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

      2. *-commutative 50.9%

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

      3. times-frac 52.5%

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

   8. Simplified 52.5%

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

   9. Taylor expanded in t around 0 50.9%

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

       1. associate-*r/ 50.9%

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

       2. *-commutative 50.9%

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

       3. associate-/r* 53.0%

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

       4. associate-*l/ 53.0%

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

       5. associate-/l* 52.6%

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

   11. Simplified 52.6%

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

   if 1.40000000000000012e-78 < t

   1. Initial program 75.4%

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

      1. associate-/r* 75.4%

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

      2. sqr-neg 75.4%

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

      3. associate-*l* 69.0%

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

      4. sqr-neg 69.0%

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

      5. associate-/r* 72.9%

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

      6. associate-+l+ 72.9%

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

      7. unpow2 72.9%

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

      8. times-frac 67.6%

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

      9. sqr-neg 67.6%

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

      10. times-frac 72.9%

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

      11. unpow2 72.9%

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

   3. Simplified 72.9%

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

   5. Taylor expanded in k around 0 66.4%

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

      1. *-commutative 66.4%

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

      2. associate-/r* 64.1%

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

   7. Simplified 64.1%

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

      1. unpow2 64.1%

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

   9. Applied egg-rr 64.1%

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

4. Final simplification 56.1%

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq 1.4 \cdot 10^{-78}:\\ \;\;\;\;\frac{\frac{2}{t}}{\frac{{k}^{4}}{{\ell}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{{\ell}^{2}}{{t}^{3}}}{k \cdot k}\\ \end{array} \]
5. Add Preprocessing

Alternative 16: 51.6% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 t_m) (* (pow l 2.0) (pow k -4.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in t around 0 63.7%

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

   1. times-frac 63.2%

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

   2. *-commutative 63.2%

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

   3. associate-/l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in k around 0 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-commutative 52.1%

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

   3. times-frac 52.9%

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

8. Simplified 52.9%

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

   1. expm1-log1p-u 52.8%

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

   2. expm1-udef 53.1%

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

   3. div-inv 53.1%

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

   4. pow-flip 53.1%

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

   5. metadata-eval 53.1%

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

10. Applied egg-rr 53.1%

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

    1. expm1-def 52.8%

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

    2. expm1-log1p 52.8%

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

12. Simplified 52.8%

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

13. Final simplification 52.8%

    \[\leadsto \frac{2}{t} \cdot \left({\ell}^{2} \cdot {k}^{-4}\right) \]
14. Add Preprocessing

Alternative 17: 51.6% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (* (/ 2.0 t_m) (/ (pow l 2.0) (pow k 4.0)))))

C

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

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in t around 0 63.7%

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

   1. times-frac 63.2%

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

   2. *-commutative 63.2%

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

   3. associate-/l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in k around 0 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-commutative 52.1%

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

   3. times-frac 52.9%

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

8. Simplified 52.9%

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

9. Final simplification 52.9%

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

Alternative 18: 51.6% accurate, 2.0× speedup

Math

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

FPCore

t_m = (fabs.f64 t)
t_s = (copysign.f64 1 t)
(FPCore (t_s t_m l k)
 :precision binary64
 (* t_s (/ (/ 2.0 t_m) (/ (pow k 4.0) (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) {
	return t_s * ((2.0 / t_m) / (pow(k, 4.0) / pow(l, 2.0)));
}

Fortran

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

Java

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

Python

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

Julia

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

MATLAB

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

Wolfram

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

Derivation

1. Initial program 53.2%

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

3. Taylor expanded in t around 0 63.7%

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

   1. times-frac 63.2%

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

   2. *-commutative 63.2%

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

   3. associate-/l* 63.3%

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

5. Simplified 63.3%

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

6. Taylor expanded in k around 0 52.1%

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

   1. associate-*r/ 52.1%

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

   2. *-commutative 52.1%

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

   3. times-frac 52.9%

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

8. Simplified 52.9%

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

9. Taylor expanded in t around 0 52.1%

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

    1. associate-*r/ 52.1%

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

    2. *-commutative 52.1%

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

    3. associate-/r* 53.1%

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

    4. associate-*l/ 53.1%

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

    5. associate-/l* 52.9%

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

11. Simplified 52.9%

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

12. Final simplification 52.9%

    \[\leadsto \frac{\frac{2}{t}}{\frac{{k}^{4}}{{\ell}^{2}}} \]
13. Add Preprocessing

Reproduce

herbie shell --seed 2024031 
(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))))