Toniolo and Linder, Equation (10-)

Percentage Accurate: 34.4% → 94.8%

Time: 24.2s

Precision: binary64

Cost: 20228

• 34.5% of points produce a very large (infinite) output. You may want to add a precondition.

• could not determine a ground truth

  1. t = 4.94066e-324
  2. l = -2
  3. k = 0.0

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0.5:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{-2}\right) \cdot \cos k}{t}\\ \end{array} \]

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))))

↓

(FPCore (t l k)
 :precision binary64
 (if (<= (* l l) 0.5)
   (/ (* l (/ (/ 2.0 k) (* k t))) (* (/ (sin k) l) (tan k)))
   (/ (* (* 2.0 (pow (* (sin k) (/ k l)) -2.0)) (cos k)) t)))

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));
}

↓

double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.5) {
		tmp = (l * ((2.0 / k) / (k * t))) / ((sin(k) / l) * tan(k));
	} else {
		tmp = ((2.0 * pow((sin(k) * (k / l)), -2.0)) * cos(k)) / t;
	}
	return tmp;
}

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

↓

real(8) function code(t, l, k)
    real(8), intent (in) :: t
    real(8), intent (in) :: l
    real(8), intent (in) :: k
    real(8) :: tmp
    if ((l * l) <= 0.5d0) then
        tmp = (l * ((2.0d0 / k) / (k * t))) / ((sin(k) / l) * tan(k))
    else
        tmp = ((2.0d0 * ((sin(k) * (k / l)) ** (-2.0d0))) * cos(k)) / t
    end if
    code = tmp
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));
}

↓

public static double code(double t, double l, double k) {
	double tmp;
	if ((l * l) <= 0.5) {
		tmp = (l * ((2.0 / k) / (k * t))) / ((Math.sin(k) / l) * Math.tan(k));
	} else {
		tmp = ((2.0 * Math.pow((Math.sin(k) * (k / l)), -2.0)) * Math.cos(k)) / t;
	}
	return tmp;
}

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))

↓

def code(t, l, k):
	tmp = 0
	if (l * l) <= 0.5:
		tmp = (l * ((2.0 / k) / (k * t))) / ((math.sin(k) / l) * math.tan(k))
	else:
		tmp = ((2.0 * math.pow((math.sin(k) * (k / l)), -2.0)) * math.cos(k)) / t
	return tmp

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

↓

function code(t, l, k)
	tmp = 0.0
	if (Float64(l * l) <= 0.5)
		tmp = Float64(Float64(l * Float64(Float64(2.0 / k) / Float64(k * t))) / Float64(Float64(sin(k) / l) * tan(k)));
	else
		tmp = Float64(Float64(Float64(2.0 * (Float64(sin(k) * Float64(k / l)) ^ -2.0)) * cos(k)) / t);
	end
	return tmp
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

↓

function tmp_2 = code(t, l, k)
	tmp = 0.0;
	if ((l * l) <= 0.5)
		tmp = (l * ((2.0 / k) / (k * t))) / ((sin(k) / l) * tan(k));
	else
		tmp = ((2.0 * ((sin(k) * (k / l)) ^ -2.0)) * cos(k)) / t;
	end
	tmp_2 = tmp;
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]

↓

code[t_, l_, k_] := If[LessEqual[N[(l * l), $MachinePrecision], 0.5], N[(N[(l * N[(N[(2.0 / k), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(N[(N[Sin[k], $MachinePrecision] / l), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(2.0 * N[Power[N[(N[Sin[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision] * N[Cos[k], $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 0.5

   1. Initial program 29.0%

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

   2. Simplified 49.2%

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

      [Start] 29.0%	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 29.0%	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 29.0%	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 29.0%	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 29.0%	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 29.0%	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 30.5%	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 30.5%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 42.3%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 42.3%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 42.3%	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 49.2%	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 87.5%

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

   4. Simplified 87.5%

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

      [Start] 87.5%	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 87.5%	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 87.6%

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

      [Start] 87.5%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      clear-num [=>] 87.5%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \left(\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \frac{\ell}{\tan k}\right) \]
      frac-times [=>] 87.6%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell} \cdot \tan k}} \]
      *-un-lft-identity [<=] 87.6%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\color{blue}{\ell}}{\frac{\sin k}{\ell} \cdot \tan k} \]

   6. Applied egg-rr 62.8%

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

      [Start] 87.6%	\[ \frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k} \]
      expm1-log1p-u [=>] 69.4%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\right)\right)} \]
      expm1-udef [=>] 62.2%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(k \cdot k\right) \cdot t} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\right)} - 1} \]
      associate-*l* [=>] 62.8%	\[ e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\right)} - 1 \]

   7. Simplified 94.8%

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

      [Start] 62.8%	\[ e^{\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\right)} - 1 \]
      expm1-def [=>] 72.3%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\right)\right)} \]
      expm1-log1p [=>] 91.9%	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}} \]
      associate-*r/ [=>] 94.8%	\[ \color{blue}{\frac{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \ell}{\frac{\sin k}{\ell} \cdot \tan k}} \]
      *-commutative [=>] 94.8%	\[ \frac{\color{blue}{\ell \cdot \frac{2}{k \cdot \left(k \cdot t\right)}}}{\frac{\sin k}{\ell} \cdot \tan k} \]
      associate-/r* [=>] 94.8%	\[ \frac{\ell \cdot \color{blue}{\frac{\frac{2}{k}}{k \cdot t}}}{\frac{\sin k}{\ell} \cdot \tan k} \]

   if 0.5 < (*.f64 l l)

   1. Initial program 29.3%

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

   2. Taylor expanded in t around 0 68.5%

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

   3. Simplified 68.6%

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

      [Start] 68.5%	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      associate-/r* [=>] 68.6%	\[ \frac{2}{\color{blue}{\frac{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}{{\ell}^{2}}}} \]
      unpow2 [=>] 68.6%	\[ \frac{2}{\frac{\frac{\color{blue}{\left(k \cdot k\right)} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k}}{{\ell}^{2}}} \]
      *-commutative [=>] 68.6%	\[ \frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}{\cos k}}{{\ell}^{2}}} \]
      unpow2 [=>] 68.6%	\[ \frac{2}{\frac{\frac{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}{\cos k}}{\color{blue}{\ell \cdot \ell}}} \]

   4. Taylor expanded in k around inf 68.5%

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

   5. Simplified 95.2%

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

      [Start] 68.5%	\[ \frac{2}{\frac{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)}{\cos k \cdot {\ell}^{2}}} \]
      *-commutative [=>] 68.5%	\[ \frac{2}{\frac{{k}^{2} \cdot \color{blue}{\left(t \cdot {\sin k}^{2}\right)}}{\cos k \cdot {\ell}^{2}}} \]
      *-commutative [<=] 68.5%	\[ \frac{2}{\frac{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}{\color{blue}{{\ell}^{2} \cdot \cos k}}} \]
      times-frac [=>] 71.6%	\[ \frac{2}{\color{blue}{\frac{{k}^{2}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}}} \]
      unpow2 [=>] 71.6%	\[ \frac{2}{\frac{\color{blue}{k \cdot k}}{{\ell}^{2}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]
      unpow2 [=>] 71.6%	\[ \frac{2}{\frac{k \cdot k}{\color{blue}{\ell \cdot \ell}} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]
      times-frac [=>] 95.2%	\[ \frac{2}{\color{blue}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)} \cdot \frac{t \cdot {\sin k}^{2}}{\cos k}} \]
      *-commutative [<=] 95.2%	\[ \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{\color{blue}{{\sin k}^{2} \cdot t}}{\cos k}} \]
      associate-/l* [=>] 95.2%	\[ \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \color{blue}{\frac{{\sin k}^{2}}{\frac{\cos k}{t}}}} \]

   6. Applied egg-rr 30.9%

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

      [Start] 95.2%	\[ \frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\frac{\cos k}{t}}} \]
      expm1-log1p-u [=>] 43.8%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\frac{\cos k}{t}}}\right)\right)} \]
      expm1-udef [=>] 30.9%	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{{\sin k}^{2}}{\frac{\cos k}{t}}}\right)} - 1} \]
      associate-*r/ [=>] 30.9%	\[ e^{\mathsf{log1p}\left(\frac{2}{\color{blue}{\frac{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot {\sin k}^{2}}{\frac{\cos k}{t}}}}\right)} - 1 \]
      pow2 [=>] 30.9%	\[ e^{\mathsf{log1p}\left(\frac{2}{\frac{\color{blue}{{\left(\frac{k}{\ell}\right)}^{2}} \cdot {\sin k}^{2}}{\frac{\cos k}{t}}}\right)} - 1 \]
      pow-prod-down [=>] 30.9%	\[ e^{\mathsf{log1p}\left(\frac{2}{\frac{\color{blue}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}}{\frac{\cos k}{t}}}\right)} - 1 \]

   7. Simplified 95.5%

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

      [Start] 30.9%	\[ e^{\mathsf{log1p}\left(\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}{\frac{\cos k}{t}}}\right)} - 1 \]
      expm1-def [=>] 43.7%	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}{\frac{\cos k}{t}}}\right)\right)} \]
      expm1-log1p [=>] 95.1%	\[ \color{blue}{\frac{2}{\frac{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}}{\frac{\cos k}{t}}}} \]
      associate-/r/ [=>] 95.5%	\[ \color{blue}{\frac{2}{{\left(\frac{k}{\ell} \cdot \sin k\right)}^{2}} \cdot \frac{\cos k}{t}} \]
      *-commutative [=>] 95.5%	\[ \frac{2}{{\color{blue}{\left(\sin k \cdot \frac{k}{\ell}\right)}}^{2}} \cdot \frac{\cos k}{t} \]

   8. Applied egg-rr 95.8%

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

      [Start] 95.5%	\[ \frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \frac{\cos k}{t} \]
      associate-*r/ [=>] 95.5%	\[ \color{blue}{\frac{\frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}} \cdot \cos k}{t}} \]
      div-inv [=>] 95.5%	\[ \frac{\color{blue}{\left(2 \cdot \frac{1}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\right)} \cdot \cos k}{t} \]
      pow-flip [=>] 95.8%	\[ \frac{\left(2 \cdot \color{blue}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{\left(-2\right)}}\right) \cdot \cos k}{t} \]
      metadata-eval [=>] 95.8%	\[ \frac{\left(2 \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{\color{blue}{-2}}\right) \cdot \cos k}{t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 95.3%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0.5:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{-2}\right) \cdot \cos k}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	94.8%
Cost	20228

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0.5:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(2 \cdot {\left(\sin k \cdot \frac{k}{\ell}\right)}^{-2}\right) \cdot \cos k}{t}\\ \end{array} \]

Alternative 2
Accuracy	95.0%
Cost	20228

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+66}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 3
Accuracy	95.1%
Cost	20228

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+66}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2 \cdot \frac{\cos k}{t}}{{\left(\sin k \cdot \frac{k}{\ell}\right)}^{2}}\\ \end{array} \]

Alternative 4
Accuracy	94.6%
Cost	14404

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+70}:\\ \;\;\;\;\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \frac{0.5 - \frac{\cos \left(k + k\right)}{2}}{\frac{\cos k}{t}}}\\ \end{array} \]

Alternative 5
Accuracy	80.9%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k \cdot k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\\ \end{array} \]

Alternative 6
Accuracy	80.9%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k \cdot k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}\\ \end{array} \]

Alternative 7
Accuracy	80.8%
Cost	14020

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 0:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k \cdot k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\sin k \cdot \frac{\tan k}{\ell}}\\ \end{array} \]

Alternative 8
Accuracy	73.6%
Cost	13828

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\frac{\cos k}{t} \cdot \frac{2}{{\left(\frac{k \cdot k}{\ell}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} + \frac{\ell \cdot \ell}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 9
Accuracy	88.4%
Cost	13760

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

Alternative 10
Accuracy	88.9%
Cost	13760

\[\frac{\ell \cdot \frac{\frac{2}{k}}{k \cdot t}}{\frac{\sin k}{\ell} \cdot \tan k} \]

Alternative 11
Accuracy	72.9%
Cost	8260

\[\begin{array}{l} t_1 := k \cdot \left(k \cdot t\right)\\ \mathbf{if}\;\ell \cdot \ell \leq 4 \cdot 10^{-287}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot t_1}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell \cdot \ell}{t_1} + \frac{\ell \cdot \ell}{t} \cdot 0.3333333333333333\right)\right)\\ \end{array} \]

Alternative 12
Accuracy	71.9%
Cost	7748

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{-147}:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\frac{\ell \cdot \ell}{t}}{k \cdot k}\right)\\ \end{array} \]

Alternative 13
Accuracy	71.3%
Cost	7044

\[\begin{array}{l} \mathbf{if}\;k \leq 2000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{-0.3333333333333333}{t} \cdot {\left(\frac{\ell}{k}\right)}^{2}\\ \end{array} \]

Alternative 14
Accuracy	69.5%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq 2000000:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{-0.3333333333333333} \cdot \frac{k \cdot t}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 15
Accuracy	70.7%
Cost	1092

\[\begin{array}{l} \mathbf{if}\;k \leq 2000000:\\ \;\;\;\;\frac{2}{\left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot \left(k \cdot t\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{1}{\frac{k}{-0.3333333333333333} \cdot \frac{k \cdot t}{\ell \cdot \ell}}\\ \end{array} \]

Alternative 16
Accuracy	33.5%
Cost	704

\[\frac{\ell \cdot \ell}{k \cdot \left(k \cdot t\right)} \cdot -0.3333333333333333 \]

Alternative 17
Accuracy	33.7%
Cost	704

\[\frac{-0.3333333333333333}{k} \cdot \frac{\ell \cdot \ell}{k \cdot t} \]

Reproduce

herbie shell --seed 2023165 
(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))))