Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 18.0

Time: 33.7s

Precision: binary64

Cost: 107468

Math

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

↓

\[\begin{array}{l} t_1 := {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \tan k \cdot 0.5\\ t_3 := \frac{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell} \cdot t_2}}{\sin k}}{2 + t_1}\\ t_4 := \left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + t_1\right) + 1\right)\\ t_5 := \frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}\\ \mathbf{if}\;t_4 \leq -2 \cdot 10^{-286}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t_4 \leq 0:\\ \;\;\;\;\frac{\frac{t_5}{\sin k}}{t_2}\\ \mathbf{elif}\;t_4 \leq \infty:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{\sin k}{t_5}}\\ \end{array} \]

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
 (let* ((t_1 (pow (/ k t) 2.0))
        (t_2 (* (tan k) 0.5))
        (t_3 (/ (/ (/ l (* (/ (pow t 3.0) l) t_2)) (sin k)) (+ 2.0 t_1)))
        (t_4
         (*
          (* (* (/ (pow t 3.0) (* l l)) (sin k)) (tan k))
          (+ (+ 1.0 t_1) 1.0)))
        (t_5 (/ l (* (pow k 2.0) (/ t l)))))
   (if (<= t_4 -2e-286)
     t_3
     (if (<= t_4 0.0)
       (/ (/ t_5 (sin k)) t_2)
       (if (<= t_4 INFINITY) t_3 (/ (/ 2.0 (tan k)) (/ (sin k) t_5)))))))

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 t_1 = pow((k / t), 2.0);
	double t_2 = tan(k) * 0.5;
	double t_3 = ((l / ((pow(t, 3.0) / l) * t_2)) / sin(k)) / (2.0 + t_1);
	double t_4 = (((pow(t, 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + t_1) + 1.0);
	double t_5 = l / (pow(k, 2.0) * (t / l));
	double tmp;
	if (t_4 <= -2e-286) {
		tmp = t_3;
	} else if (t_4 <= 0.0) {
		tmp = (t_5 / sin(k)) / t_2;
	} else if (t_4 <= ((double) INFINITY)) {
		tmp = t_3;
	} else {
		tmp = (2.0 / tan(k)) / (sin(k) / t_5);
	}
	return tmp;
}

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 t_1 = Math.pow((k / t), 2.0);
	double t_2 = Math.tan(k) * 0.5;
	double t_3 = ((l / ((Math.pow(t, 3.0) / l) * t_2)) / Math.sin(k)) / (2.0 + t_1);
	double t_4 = (((Math.pow(t, 3.0) / (l * l)) * Math.sin(k)) * Math.tan(k)) * ((1.0 + t_1) + 1.0);
	double t_5 = l / (Math.pow(k, 2.0) * (t / l));
	double tmp;
	if (t_4 <= -2e-286) {
		tmp = t_3;
	} else if (t_4 <= 0.0) {
		tmp = (t_5 / Math.sin(k)) / t_2;
	} else if (t_4 <= Double.POSITIVE_INFINITY) {
		tmp = t_3;
	} else {
		tmp = (2.0 / Math.tan(k)) / (Math.sin(k) / t_5);
	}
	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):
	t_1 = math.pow((k / t), 2.0)
	t_2 = math.tan(k) * 0.5
	t_3 = ((l / ((math.pow(t, 3.0) / l) * t_2)) / math.sin(k)) / (2.0 + t_1)
	t_4 = (((math.pow(t, 3.0) / (l * l)) * math.sin(k)) * math.tan(k)) * ((1.0 + t_1) + 1.0)
	t_5 = l / (math.pow(k, 2.0) * (t / l))
	tmp = 0
	if t_4 <= -2e-286:
		tmp = t_3
	elif t_4 <= 0.0:
		tmp = (t_5 / math.sin(k)) / t_2
	elif t_4 <= math.inf:
		tmp = t_3
	else:
		tmp = (2.0 / math.tan(k)) / (math.sin(k) / t_5)
	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)
	t_1 = Float64(k / t) ^ 2.0
	t_2 = Float64(tan(k) * 0.5)
	t_3 = Float64(Float64(Float64(l / Float64(Float64((t ^ 3.0) / l) * t_2)) / sin(k)) / Float64(2.0 + t_1))
	t_4 = Float64(Float64(Float64(Float64((t ^ 3.0) / Float64(l * l)) * sin(k)) * tan(k)) * Float64(Float64(1.0 + t_1) + 1.0))
	t_5 = Float64(l / Float64((k ^ 2.0) * Float64(t / l)))
	tmp = 0.0
	if (t_4 <= -2e-286)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = Float64(Float64(t_5 / sin(k)) / t_2);
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = Float64(Float64(2.0 / tan(k)) / Float64(sin(k) / t_5));
	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)
	t_1 = (k / t) ^ 2.0;
	t_2 = tan(k) * 0.5;
	t_3 = ((l / (((t ^ 3.0) / l) * t_2)) / sin(k)) / (2.0 + t_1);
	t_4 = ((((t ^ 3.0) / (l * l)) * sin(k)) * tan(k)) * ((1.0 + t_1) + 1.0);
	t_5 = l / ((k ^ 2.0) * (t / l));
	tmp = 0.0;
	if (t_4 <= -2e-286)
		tmp = t_3;
	elseif (t_4 <= 0.0)
		tmp = (t_5 / sin(k)) / t_2;
	elseif (t_4 <= Inf)
		tmp = t_3;
	else
		tmp = (2.0 / tan(k)) / (sin(k) / t_5);
	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_] := Block[{t$95$1 = N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(N[Tan[k], $MachinePrecision] * 0.5), $MachinePrecision]}, Block[{t$95$3 = N[(N[(N[(l / N[(N[(N[Power[t, 3.0], $MachinePrecision] / l), $MachinePrecision] * t$95$2), $MachinePrecision]), $MachinePrecision] / N[Sin[k], $MachinePrecision]), $MachinePrecision] / N[(2.0 + t$95$1), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = 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 + t$95$1), $MachinePrecision] + 1.0), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[(l / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t$95$4, -2e-286], t$95$3, If[LessEqual[t$95$4, 0.0], N[(N[(t$95$5 / N[Sin[k], $MachinePrecision]), $MachinePrecision] / t$95$2), $MachinePrecision], If[LessEqual[t$95$4, Infinity], t$95$3, N[(N[(2.0 / N[Tan[k], $MachinePrecision]), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] / t$95$5), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 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)) < -2.0000000000000001e-286 or -0.0 < (*.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)) < +inf.0

   1. Initial program 12.5

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

   2. Simplified 10.4

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

      [Start] 12.5	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 12.6	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-46 [=>] 12.6	\[ \frac{\color{blue}{\frac{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}{\tan k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 12.6	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 12.5	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}{\sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 11.9	\[ \frac{\frac{\color{blue}{\frac{\ell \cdot \ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-49 [=>] 10.4	\[ \frac{\frac{\color{blue}{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}}{\sin k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 10.4	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 10.4	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 10.4	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 10.4	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 10.4	\[ \frac{\frac{\ell \cdot \frac{\ell}{\frac{{t}^{3}}{\frac{2}{\tan k}}}}{\sin k}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 9.4

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

   if -2.0000000000000001e-286 < (*.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)) < -0.0

   1. Initial program 60.4

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

   2. Simplified 38.5

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

      [Start] 60.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 60.4	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-2 [=>] 60.4	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 60.4	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-2 [=>] 60.4	\[ \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 62.5	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 37.7	\[ \frac{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 38.5	\[ \frac{\color{blue}{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\tan k}}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 38.5	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 21.0

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

   4. Taylor expanded in t around 0 31.2

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

   5. Simplified 24.3

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

      [Start] 31.2	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}} \]
      rational.json-simplify-2 [=>] 31.2	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}} \]
      rational.json-simplify-49 [=>] 24.3	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot \frac{t}{\ell}}} \]

   6. Applied egg-rr 24.3

      \[\leadsto \color{blue}{\frac{\frac{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}{\sin k}}{\tan k \cdot 0.5}} \]

   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 64.0

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

   2. Simplified 59.8

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

      [Start] 64.0	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
      rational.json-simplify-46 [=>] 64.0	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1}} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 64.0	\[ \frac{\color{blue}{\frac{\frac{2}{\tan k}}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-2 [=>] 64.0	\[ \frac{\frac{\frac{2}{\tan k}}{\color{blue}{\sin k \cdot \frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 64.0	\[ \frac{\color{blue}{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\frac{{t}^{3}}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-46 [=>] 59.8	\[ \frac{\frac{\frac{\frac{2}{\tan k}}{\sin k}}{\color{blue}{\frac{\frac{{t}^{3}}{\ell}}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-61 [=>] 59.8	\[ \frac{\color{blue}{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\tan k}}{\sin k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-44 [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\color{blue}{\frac{\frac{2}{\sin k}}{\tan k}}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1} \]
      rational.json-simplify-1 [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{1 + \left(1 + {\left(\frac{k}{t}\right)}^{2}\right)}} \]
      rational.json-simplify-1 [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{1 + \color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)}} \]
      rational.json-simplify-41 [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 + 1\right)}} \]
      metadata-eval [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{2}} \]
      rational.json-simplify-1 [=>] 59.8	\[ \frac{\frac{\ell}{\frac{\frac{{t}^{3}}{\ell}}{\frac{\frac{2}{\sin k}}{\tan k}}}}{\color{blue}{2 + {\left(\frac{k}{t}\right)}^{2}}} \]

   3. Applied egg-rr 59.5

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

   4. Taylor expanded in t around 0 33.4

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

   5. Simplified 32.4

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

      [Start] 33.4	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}} \]
      rational.json-simplify-2 [=>] 33.4	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\frac{\color{blue}{t \cdot {k}^{2}}}{\ell}} \]
      rational.json-simplify-49 [=>] 32.4	\[ \frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\color{blue}{{k}^{2} \cdot \frac{t}{\ell}}} \]

   6. Applied egg-rr 32.4

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

4. Final simplification 18.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq -2 \cdot 10^{-286}:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot 0.5\right)}}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq 0:\\ \;\;\;\;\frac{\frac{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}{\sin k}}{\tan k \cdot 0.5}\\ \mathbf{elif}\;\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right) \leq \infty:\\ \;\;\;\;\frac{\frac{\frac{\ell}{\frac{{t}^{3}}{\ell} \cdot \left(\tan k \cdot 0.5\right)}}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{\sin k}{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}}\\ \end{array} \]

Alternatives

Alternative 1
Error	16.9
Cost	27080

\[\begin{array}{l} t_1 := \ell \cdot \left(\frac{2}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)\\ \mathbf{if}\;t \leq -1.45 \cdot 10^{-70}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-41}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{\sin k}{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 2
Error	16.9
Cost	27080

\[\begin{array}{l} t_1 := \frac{2}{\tan k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -9.2 \cdot 10^{-38}:\\ \;\;\;\;\frac{\ell}{t_2} \cdot \left(\frac{\ell}{\sin k} \cdot \frac{t_1}{{t}^{3}}\right)\\ \mathbf{elif}\;t \leq 5.3 \cdot 10^{-39}:\\ \;\;\;\;\frac{t_1}{\frac{\sin k}{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{{t}^{3} \cdot \tan k} \cdot \frac{\ell}{\sin k \cdot t_2}\right)\\ \end{array} \]

Alternative 3
Error	19.7
Cost	20548

\[\begin{array}{l} \mathbf{if}\;t \leq -4.6 \cdot 10^{-83}:\\ \;\;\;\;\frac{\frac{\ell \cdot \left(2 \cdot \frac{\frac{\ell}{k}}{{t}^{3}}\right)}{\sin k}}{2 + {\left(\frac{k}{t}\right)}^{2}}\\ \mathbf{elif}\;t \leq 3.9 \cdot 10^{-22}:\\ \;\;\;\;\frac{\frac{2}{\tan k}}{\frac{\sin k}{\frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}}}\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot \frac{\frac{2}{k}}{\frac{2}{\ell} \cdot \left(k \cdot {t}^{3}\right)}\\ \end{array} \]

Alternative 4
Error	20.6
Cost	20360

\[\begin{array}{l} t_1 := \ell \cdot \frac{\ell}{t \cdot \left({k}^{2} \cdot \left(\sin k \cdot \left(0.5 \cdot \tan k\right)\right)\right)}\\ \mathbf{if}\;k \leq -7 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 6.4 \cdot 10^{-18}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 5
Error	20.9
Cost	20360

\[\begin{array}{l} t_1 := \frac{\ell}{\sin k} \cdot \left(\frac{2}{\tan k} \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \mathbf{if}\;k \leq -7.2 \cdot 10^{-27}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 6
Error	20.9
Cost	20360

\[\begin{array}{l} \mathbf{if}\;k \leq -3.5 \cdot 10^{-27}:\\ \;\;\;\;\frac{\frac{2}{\sin k}}{\tan k} \cdot \frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}}\\ \mathbf{elif}\;k \leq 1.05 \cdot 10^{-50}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(\frac{2}{\tan k} \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \]

Alternative 7
Error	20.9
Cost	20360

\[\begin{array}{l} t_1 := \frac{2}{\tan k}\\ \mathbf{if}\;k \leq -5.6 \cdot 10^{-27}:\\ \;\;\;\;\frac{t_1}{\frac{\sin k}{\frac{\ell}{\frac{{k}^{2} \cdot t}{\ell}}}}\\ \mathbf{elif}\;k \leq 9.2 \cdot 10^{-51}:\\ \;\;\;\;\frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{\sin k} \cdot \left(t_1 \cdot \frac{\ell}{t \cdot {k}^{2}}\right)\\ \end{array} \]

Alternative 8
Error	23.9
Cost	14408

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -2.3 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 4.2 \cdot 10^{-81}:\\ \;\;\;\;\left(2 \cdot \frac{1}{{k}^{2}} - 0.3333333333333333\right) \cdot \frac{\ell}{\frac{{k}^{2} \cdot \frac{2}{\ell}}{\frac{2}{t}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 9
Error	23.9
Cost	14280

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-82}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.8 \cdot 10^{-81}:\\ \;\;\;\;\frac{\frac{2 \cdot \frac{1}{{k}^{2}} + -0.3333333333333333}{\frac{1}{\ell}}}{{k}^{2} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 10
Error	23.9
Cost	14152

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -7.9 \cdot 10^{-84}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.22 \cdot 10^{-79}:\\ \;\;\;\;\left(2 \cdot \frac{1}{{k}^{2}} - 0.3333333333333333\right) \cdot \frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 11
Error	25.6
Cost	13896

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.45 \cdot 10^{-82}:\\ \;\;\;\;2 \cdot \left(\frac{{\ell}^{2}}{k \cdot t} \cdot \frac{1}{{k}^{3}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 12
Error	24.2
Cost	13896

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -2.35 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.5 \cdot 10^{-80}:\\ \;\;\;\;\frac{2}{{k}^{2}} \cdot \frac{\ell}{{k}^{2} \cdot \frac{t}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 13
Error	25.9
Cost	13768

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -2.5 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-83}:\\ \;\;\;\;2 \cdot \frac{\frac{{\ell}^{2}}{k}}{t \cdot {k}^{3}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 14
Error	26.4
Cost	13640

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -4.1 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 2 \cdot 10^{-81}:\\ \;\;\;\;-0.3333333333333333 \cdot \frac{\frac{{\ell}^{2}}{{k}^{2}}}{t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 15
Error	26.1
Cost	13640

\[\begin{array}{l} t_1 := \frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}}\\ \mathbf{if}\;t \leq -3.1 \cdot 10^{-83}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 3.2 \cdot 10^{-80}:\\ \;\;\;\;2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

Alternative 16
Error	32.7
Cost	7040

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

Alternative 17
Error	31.8
Cost	7040

\[\ell \cdot \frac{\frac{\ell}{k}}{k \cdot {t}^{3}} \]

Alternative 18
Error	32.5
Cost	7040

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

Alternative 19
Error	31.7
Cost	7040

\[\frac{\frac{\ell}{k}}{k \cdot \frac{{t}^{3}}{\ell}} \]

Reproduce

herbie shell --seed 2023064 
(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))))