Toniolo and Linder, Equation (10-)

Average Accuracy: 16.3% → 61.7%

Time: 29.5s

Precision: binary64

Cost: 14020

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\tan k}\right)}{t}\\ \end{array} \]

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) 2e+307)
   (* (/ (* 2.0 (/ l k)) (tan k)) (/ (/ l (sin k)) (* k t)))
   (/ (* (/ l (* k (sin k))) (* 2.0 (/ (/ l k) (tan 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) <= 2e+307) {
		tmp = ((2.0 * (l / k)) / tan(k)) * ((l / sin(k)) / (k * t));
	} else {
		tmp = ((l / (k * sin(k))) * (2.0 * ((l / k) / tan(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) <= 2d+307) then
        tmp = ((2.0d0 * (l / k)) / tan(k)) * ((l / sin(k)) / (k * t))
    else
        tmp = ((l / (k * sin(k))) * (2.0d0 * ((l / k) / tan(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) <= 2e+307) {
		tmp = ((2.0 * (l / k)) / Math.tan(k)) * ((l / Math.sin(k)) / (k * t));
	} else {
		tmp = ((l / (k * Math.sin(k))) * (2.0 * ((l / k) / Math.tan(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) <= 2e+307:
		tmp = ((2.0 * (l / k)) / math.tan(k)) * ((l / math.sin(k)) / (k * t))
	else:
		tmp = ((l / (k * math.sin(k))) * (2.0 * ((l / k) / math.tan(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) <= 2e+307)
		tmp = Float64(Float64(Float64(2.0 * Float64(l / k)) / tan(k)) * Float64(Float64(l / sin(k)) / Float64(k * t)));
	else
		tmp = Float64(Float64(Float64(l / Float64(k * sin(k))) * Float64(2.0 * Float64(Float64(l / k) / tan(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) <= 2e+307)
		tmp = ((2.0 * (l / k)) / tan(k)) * ((l / sin(k)) / (k * t));
	else
		tmp = ((l / (k * sin(k))) * (2.0 * ((l / k) / tan(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], 2e+307], N[(N[(N[(2.0 * N[(l / k), $MachinePrecision]), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[(N[(l / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(k * t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(N[(l / N[(k * N[Sin[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[(N[(l / k), $MachinePrecision] / N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 1.99999999999999997e307

   1. Initial program 22.7%

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

   2. Simplified 35.6%

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

      [Start] 22.7	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 22.7	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 22.7	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 22.6	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 22.6	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 22.6	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 22.7	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 22.7	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 35.1	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 35.1	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 35.1	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 35.6	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 64.0%

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

   4. Simplified 69.0%

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

      [Start] 64.0	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 64.0	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l* [=>] 69.0	\[ \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 49.9%

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

      [Start] 69.0	\[ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      expm1-log1p-u [=>] 62.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]
      expm1-udef [=>] 49.9	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]
      associate-*l/ [=>] 49.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}}\right)} - 1 \]
      *-un-lft-identity [=>] 49.9	\[ e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)} - 1 \]
      times-frac [=>] 49.9	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{1} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}}\right)} - 1 \]
      metadata-eval [=>] 49.9	\[ e^{\mathsf{log1p}\left(\color{blue}{2} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      clear-num [=>] 49.9	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      frac-times [=>] 49.9	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell} \cdot \tan k}}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      *-un-lft-identity [<=] 49.9	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\frac{\color{blue}{\ell}}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]

   6. Simplified 69.9%

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

      [Start] 49.9	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      expm1-def [=>] 62.2	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)\right)} \]
      expm1-log1p [=>] 69.0	\[ \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 69.0	\[ \color{blue}{\frac{2 \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}} \]
      associate-*l/ [<=] 69.0	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}} \]
      associate-/r* [=>] 69.2	\[ \color{blue}{\frac{\frac{2}{k}}{k \cdot t}} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k} \]
      associate-/r* [=>] 68.8	\[ \frac{\frac{2}{k}}{k \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\tan k}} \]
      times-frac [<=] 69.4	\[ \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\left(k \cdot t\right) \cdot \tan k}} \]
      *-commutative [<=] 69.4	\[ \frac{\frac{2}{k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\tan k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 69.9	\[ \frac{\color{blue}{\frac{\frac{2}{k} \cdot \ell}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]

   7. Applied egg-rr 74.9%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t}} \]
      Proof

      [Start] 69.9	\[ \frac{\frac{\frac{2}{k} \cdot \ell}{\frac{\sin k}{\ell}}}{\tan k \cdot \left(k \cdot t\right)} \]
      div-inv [=>] 69.9	\[ \frac{\color{blue}{\left(\frac{2}{k} \cdot \ell\right) \cdot \frac{1}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]
      times-frac [=>] 74.8	\[ \color{blue}{\frac{\frac{2}{k} \cdot \ell}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t}} \]
      associate-*l/ [=>] 74.9	\[ \frac{\color{blue}{\frac{2 \cdot \ell}{k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      *-un-lft-identity [=>] 74.9	\[ \frac{\frac{2 \cdot \ell}{\color{blue}{1 \cdot k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      times-frac [=>] 74.9	\[ \frac{\color{blue}{\frac{2}{1} \cdot \frac{\ell}{k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      metadata-eval [=>] 74.9	\[ \frac{\color{blue}{2} \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      clear-num [<=] 74.9	\[ \frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k}}}{k \cdot t} \]

   if 1.99999999999999997e307 < (*.f64 l l)

   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. Simplified 0.0%

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

      [Start] 0.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\sin k \cdot \tan k\right)\right)} \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)\right)}} \]
      associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}} \]
      associate-/r/ [=>] 0.0	\[ \frac{\color{blue}{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}}{\left(\sin k \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      *-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}} \cdot \left(\ell \cdot \ell\right)}{\color{blue}{\left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      times-frac [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{{t}^{3}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k}} \]
      +-commutative [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      associate--l+ [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      metadata-eval [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      +-rgt-identity [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \cdot \frac{\ell \cdot \ell}{\sin k \cdot \tan k} \]
      times-frac [=>] 0.0	\[ \frac{\frac{2}{{t}^{3}}}{{\left(\frac{k}{t}\right)}^{2}} \cdot \color{blue}{\left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)} \]

   3. Taylor expanded in t around 0 0.0%

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

   4. Simplified 0.0%

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

      [Start] 0.0	\[ \frac{2}{{k}^{2} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      unpow2 [=>] 0.0	\[ \frac{2}{\color{blue}{\left(k \cdot k\right)} \cdot t} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      associate-*l* [=>] 0.0	\[ \frac{2}{\color{blue}{k \cdot \left(k \cdot t\right)}} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]

   5. Applied egg-rr 0.0%

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

      [Start] 0.0	\[ \frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right) \]
      expm1-log1p-u [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)\right)} \]
      expm1-udef [=>] 0.0	\[ \color{blue}{e^{\mathsf{log1p}\left(\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)\right)} - 1} \]
      associate-*l/ [=>] 0.0	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{k \cdot \left(k \cdot t\right)}}\right)} - 1 \]
      *-un-lft-identity [=>] 0.0	\[ e^{\mathsf{log1p}\left(\frac{2 \cdot \left(\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}\right)}{\color{blue}{1 \cdot \left(k \cdot \left(k \cdot t\right)\right)}}\right)} - 1 \]
      times-frac [=>] 0.0	\[ e^{\mathsf{log1p}\left(\color{blue}{\frac{2}{1} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}}\right)} - 1 \]
      metadata-eval [=>] 0.0	\[ e^{\mathsf{log1p}\left(\color{blue}{2} \cdot \frac{\frac{\ell}{\sin k} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      clear-num [=>] 0.0	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\color{blue}{\frac{1}{\frac{\sin k}{\ell}}} \cdot \frac{\ell}{\tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      frac-times [=>] 0.0	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\color{blue}{\frac{1 \cdot \ell}{\frac{\sin k}{\ell} \cdot \tan k}}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      *-un-lft-identity [<=] 0.0	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\frac{\color{blue}{\ell}}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]

   6. Simplified 22.2%

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

      [Start] 0.0	\[ e^{\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)} - 1 \]
      expm1-def [=>] 0.0	\[ \color{blue}{\mathsf{expm1}\left(\mathsf{log1p}\left(2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}\right)\right)} \]
      expm1-log1p [=>] 0.0	\[ \color{blue}{2 \cdot \frac{\frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 0.0	\[ \color{blue}{\frac{2 \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}}{k \cdot \left(k \cdot t\right)}} \]
      associate-*l/ [<=] 0.0	\[ \color{blue}{\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k}} \]
      associate-/r* [=>] 0.0	\[ \color{blue}{\frac{\frac{2}{k}}{k \cdot t}} \cdot \frac{\ell}{\frac{\sin k}{\ell} \cdot \tan k} \]
      associate-/r* [=>] 0.0	\[ \frac{\frac{2}{k}}{k \cdot t} \cdot \color{blue}{\frac{\frac{\ell}{\frac{\sin k}{\ell}}}{\tan k}} \]
      times-frac [<=] 0.0	\[ \color{blue}{\frac{\frac{2}{k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\left(k \cdot t\right) \cdot \tan k}} \]
      *-commutative [<=] 0.0	\[ \frac{\frac{2}{k} \cdot \frac{\ell}{\frac{\sin k}{\ell}}}{\color{blue}{\tan k \cdot \left(k \cdot t\right)}} \]
      associate-*r/ [=>] 22.2	\[ \frac{\color{blue}{\frac{\frac{2}{k} \cdot \ell}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]

   7. Applied egg-rr 31.4%

      \[\leadsto \color{blue}{\frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t}} \]
      Proof

      [Start] 22.2	\[ \frac{\frac{\frac{2}{k} \cdot \ell}{\frac{\sin k}{\ell}}}{\tan k \cdot \left(k \cdot t\right)} \]
      div-inv [=>] 22.2	\[ \frac{\color{blue}{\left(\frac{2}{k} \cdot \ell\right) \cdot \frac{1}{\frac{\sin k}{\ell}}}}{\tan k \cdot \left(k \cdot t\right)} \]
      times-frac [=>] 31.4	\[ \color{blue}{\frac{\frac{2}{k} \cdot \ell}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t}} \]
      associate-*l/ [=>] 31.4	\[ \frac{\color{blue}{\frac{2 \cdot \ell}{k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      *-un-lft-identity [=>] 31.4	\[ \frac{\frac{2 \cdot \ell}{\color{blue}{1 \cdot k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      times-frac [=>] 31.4	\[ \frac{\color{blue}{\frac{2}{1} \cdot \frac{\ell}{k}}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      metadata-eval [=>] 31.4	\[ \frac{\color{blue}{2} \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{1}{\frac{\sin k}{\ell}}}{k \cdot t} \]
      clear-num [<=] 31.4	\[ \frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\color{blue}{\frac{\ell}{\sin k}}}{k \cdot t} \]

   8. Applied egg-rr 46.8%

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

      [Start] 31.4	\[ \frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t} \]
      *-commutative [=>] 31.4	\[ \color{blue}{\frac{\frac{\ell}{\sin k}}{k \cdot t} \cdot \frac{2 \cdot \frac{\ell}{k}}{\tan k}} \]
      associate-/r* [=>] 46.8	\[ \color{blue}{\frac{\frac{\frac{\ell}{\sin k}}{k}}{t}} \cdot \frac{2 \cdot \frac{\ell}{k}}{\tan k} \]
      associate-*l/ [=>] 46.8	\[ \color{blue}{\frac{\frac{\frac{\ell}{\sin k}}{k} \cdot \frac{2 \cdot \frac{\ell}{k}}{\tan k}}{t}} \]
      associate-/l/ [=>] 46.8	\[ \frac{\color{blue}{\frac{\ell}{k \cdot \sin k}} \cdot \frac{2 \cdot \frac{\ell}{k}}{\tan k}}{t} \]
      *-un-lft-identity [=>] 46.8	\[ \frac{\frac{\ell}{k \cdot \sin k} \cdot \frac{2 \cdot \frac{\ell}{k}}{\color{blue}{1 \cdot \tan k}}}{t} \]
      times-frac [=>] 46.8	\[ \frac{\frac{\ell}{k \cdot \sin k} \cdot \color{blue}{\left(\frac{2}{1} \cdot \frac{\frac{\ell}{k}}{\tan k}\right)}}{t} \]
      metadata-eval [=>] 46.8	\[ \frac{\frac{\ell}{k \cdot \sin k} \cdot \left(\color{blue}{2} \cdot \frac{\frac{\ell}{k}}{\tan k}\right)}{t} \]

3. Recombined 2 regimes into one program.

4. Final simplification 68.8%

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;\frac{2 \cdot \frac{\ell}{k}}{\tan k} \cdot \frac{\frac{\ell}{\sin k}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell}{k \cdot \sin k} \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{\tan k}\right)}{t}\\ \end{array} \]

Alternatives

Alternative 1
Accuracy	51.8%
Cost	14280

\[\begin{array}{l} t_1 := \frac{\frac{2}{k}}{\left(t \cdot \frac{k}{\frac{\ell}{k}}\right) \cdot \frac{\tan k}{\ell}}\\ \mathbf{if}\;\ell \cdot \ell \leq 10^{-317}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;\ell \cdot \ell \leq 10^{+279}:\\ \;\;\;\;\frac{2}{k} \cdot \frac{\frac{\ell \cdot \ell}{\sin k}}{t \cdot \left(k \cdot \tan k\right)}\\ \mathbf{elif}\;\ell \cdot \ell \leq 2 \cdot 10^{+307}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{-0.16666666666666666 + \frac{1}{k \cdot k}}{\frac{t}{{\left(\frac{\ell}{k}\right)}^{2}}}\\ \end{array} \]

Alternative 2
Accuracy	60.7%
Cost	13760

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

Alternative 3
Accuracy	39.9%
Cost	7360

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

Alternative 4
Accuracy	41.3%
Cost	7360

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

Alternative 5
Accuracy	39.5%
Cost	7040

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

Alternative 6
Accuracy	39.9%
Cost	1992

\[\begin{array}{l} \mathbf{if}\;k \leq -600000000:\\ \;\;\;\;\frac{\ell \cdot -0.3333333333333333}{k} \cdot \frac{\ell}{k \cdot t}\\ \mathbf{elif}\;k \leq 800:\\ \;\;\;\;2 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k} + \left(\ell \cdot \ell\right) \cdot \left(-0.16666666666666666 - \left(k \cdot k\right) \cdot 0.058333333333333334\right)}{k \cdot \left(k \cdot t\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{0.3333333333333333}{\frac{k}{\ell} \cdot \frac{-k}{\frac{\ell}{t}}}\\ \end{array} \]

Alternative 7
Accuracy	38.4%
Cost	1476

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \left(\ell \cdot \left(\ell \cdot \left(-0.16666666666666666 + \frac{1}{k \cdot k}\right)\right)\right)\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 8
Accuracy	36.2%
Cost	1220

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 5 \cdot 10^{+235}:\\ \;\;\;\;\frac{2}{k \cdot \left(k \cdot t\right)} \cdot \frac{\ell \cdot \ell}{k \cdot k}\\ \mathbf{else}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{\ell}{k} \cdot \frac{\ell}{k}}{t}\\ \end{array} \]

Alternative 9
Accuracy	36.1%
Cost	960

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

Alternative 10
Accuracy	32.0%
Cost	768

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

Alternative 11
Accuracy	31.6%
Cost	704

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

Alternative 12
Accuracy	32.0%
Cost	704

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

Alternative 13
Accuracy	32.1%
Cost	704

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

Alternative 14
Accuracy	31.9%
Cost	704

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

Alternative 15
Accuracy	32.1%
Cost	704

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

Alternative 16
Accuracy	17.5%
Cost	448

\[-0.11666666666666667 \cdot \frac{\ell}{\frac{t}{\ell}} \]

Alternative 17
Accuracy	19.8%
Cost	448

\[-0.11666666666666667 \cdot \frac{\ell \cdot \ell}{t} \]

Reproduce

herbie shell --seed 2023157 
(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))))