Average Error: 32.6 → 5.6
Time: 1.1min
Precision: binary64
Cost: 53776
\[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) + 1\right)} \]
↓
\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{2}{\mathsf{fma}\left(2, \frac{t}{\cos k} \cdot \frac{t_1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}, \frac{t}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t_1 \cdot {\left(\frac{-1}{k}\right)}^{-2}}}\right)}\\ t_3 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -5.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \tan k}\right)\\ \mathbf{elif}\;k \leq -3.7 \cdot 10^{-56}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-78}:\\ \;\;\;\;t_3 \cdot \frac{t_3}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+121}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{k}}}\\ \end{array} \]
(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 (sin k) 2.0))
        (t_2
         (/
          2.0
          (fma
           2.0
           (* (/ t (cos k)) (/ t_1 (* (/ l t) (/ l t))))
           (/ t (/ (* (cos k) (* l l)) (* t_1 (pow (/ -1.0 k) -2.0)))))))
        (t_3 (/ l (* k t))))
   (if (<= k -5.9e+144)
     (* (/ l k) (* (/ 2.0 t) (/ (/ l k) (* (sin k) (tan k)))))
     (if (<= k -3.7e-56)
       t_2
       (if (<= k 2.6e-78)
         (* t_3 (/ t_3 t))
         (if (<= k 2.5e+121)
           t_2
           (/ 2.0 (/ (* (* t (sin k)) (* (tan k) (/ k l))) (/ l k)))))))))
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(sin(k), 2.0);
	double t_2 = 2.0 / fma(2.0, ((t / cos(k)) * (t_1 / ((l / t) * (l / t)))), (t / ((cos(k) * (l * l)) / (t_1 * pow((-1.0 / k), -2.0)))));
	double t_3 = l / (k * t);
	double tmp;
	if (k <= -5.9e+144) {
		tmp = (l / k) * ((2.0 / t) * ((l / k) / (sin(k) * tan(k))));
	} else if (k <= -3.7e-56) {
		tmp = t_2;
	} else if (k <= 2.6e-78) {
		tmp = t_3 * (t_3 / t);
	} else if (k <= 2.5e+121) {
		tmp = t_2;
	} else {
		tmp = 2.0 / (((t * sin(k)) * (tan(k) * (k / l))) / (l / k));
	}
	return tmp;
}

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 = sin(k) ^ 2.0
	t_2 = Float64(2.0 / fma(2.0, Float64(Float64(t / cos(k)) * Float64(t_1 / Float64(Float64(l / t) * Float64(l / t)))), Float64(t / Float64(Float64(cos(k) * Float64(l * l)) / Float64(t_1 * (Float64(-1.0 / k) ^ -2.0))))))
	t_3 = Float64(l / Float64(k * t))
	tmp = 0.0
	if (k <= -5.9e+144)
		tmp = Float64(Float64(l / k) * Float64(Float64(2.0 / t) * Float64(Float64(l / k) / Float64(sin(k) * tan(k)))));
	elseif (k <= -3.7e-56)
		tmp = t_2;
	elseif (k <= 2.6e-78)
		tmp = Float64(t_3 * Float64(t_3 / t));
	elseif (k <= 2.5e+121)
		tmp = t_2;
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(t * sin(k)) * Float64(tan(k) * Float64(k / l))) / Float64(l / k)));
	end
	return tmp
end

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[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 / N[(2.0 * N[(N[(t / N[Cos[k], $MachinePrecision]), $MachinePrecision] * N[(t$95$1 / N[(N[(l / t), $MachinePrecision] * N[(l / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(t / N[(N[(N[Cos[k], $MachinePrecision] * N[(l * l), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[Power[N[(-1.0 / k), $MachinePrecision], -2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l / N[(k * t), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -5.9e+144], N[(N[(l / k), $MachinePrecision] * N[(N[(2.0 / t), $MachinePrecision] * N[(N[(l / k), $MachinePrecision] / N[(N[Sin[k], $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -3.7e-56], t$95$2, If[LessEqual[k, 2.6e-78], N[(t$95$3 * N[(t$95$3 / t), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, 2.5e+121], t$95$2, N[(2.0 / N[(N[(N[(t * N[Sin[k], $MachinePrecision]), $MachinePrecision] * N[(N[Tan[k], $MachinePrecision] * N[(k / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]]

\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 := {\sin k}^{2}\\
t_2 := \frac{2}{\mathsf{fma}\left(2, \frac{t}{\cos k} \cdot \frac{t_1}{\frac{\ell}{t} \cdot \frac{\ell}{t}}, \frac{t}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{t_1 \cdot {\left(\frac{-1}{k}\right)}^{-2}}}\right)}\\
t_3 := \frac{\ell}{k \cdot t}\\
\mathbf{if}\;k \leq -5.9 \cdot 10^{+144}:\\
\;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \tan k}\right)\\

\mathbf{elif}\;k \leq -3.7 \cdot 10^{-56}:\\
\;\;\;\;t_2\\

\mathbf{elif}\;k \leq 2.6 \cdot 10^{-78}:\\
\;\;\;\;t_3 \cdot \frac{t_3}{t}\\

\mathbf{elif}\;k \leq 2.5 \cdot 10^{+121}:\\
\;\;\;\;t_2\\

\mathbf{else}:\\
\;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{k}}}\\


\end{array}

Error

Derivation

  1. Split input into 4 regimes

  2. if k < -5.89999999999999988e144

    1. Initial program 35.1

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

    2. Simplified35.1

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k)))))): 0 points increase in error, 3 points decrease in error
      (/.f64 2 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))))): 3 points increase in error, 32 points decrease in error
      (/.f64 2 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

    3. Taylor expanded in t around 0 24.1

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

    4. Simplified5.5

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      Proof
      (/.f64 t (*.f64 (/.f64 l k) (/.f64 l k))): 0 points increase in error, 0 points decrease in error
      (/.f64 t (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k)))): 66 points increase in error, 17 points decrease in error
      (/.f64 t (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (/.f64 t (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t (pow.f64 k 2)) (pow.f64 l 2))): 15 points increase in error, 12 points decrease in error

    5. Applied egg-rr2.6

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

    6. Applied egg-rr2.5

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

    if -5.89999999999999988e144 < k < -3.7000000000000002e-56 or 2.6000000000000001e-78 < k < 2.50000000000000004e121

    1. Initial program 30.1

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

    2. Simplified29.9

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k)))))): 0 points increase in error, 3 points decrease in error
      (/.f64 2 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))))): 3 points increase in error, 32 points decrease in error
      (/.f64 2 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

    3. Taylor expanded in k around -inf 19.0

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

    4. Simplified8.9

      \[\leadsto \frac{2}{\color{blue}{\mathsf{fma}\left(2, \frac{t}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{t} \cdot \frac{\ell}{t}}, \frac{t}{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{{\left(\frac{-1}{k}\right)}^{-2} \cdot {\sin k}^{2}}}\right)}} \]
      Proof
      (fma.f64 2 (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (*.f64 (/.f64 l t) (/.f64 l t)))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 t t))))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 22 points increase in error, 2 points decrease in error
      (fma.f64 2 (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 t t)))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 (/.f64 t (cos.f64 k)) (/.f64 (pow.f64 (sin.f64 k) 2) (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 t 2))))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (*.f64 (/.f64 t (cos.f64 k)) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2)) (pow.f64 l 2)))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 2 points increase in error, 6 points decrease in error
      (fma.f64 2 (Rewrite<= times-frac_binary64 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2)))) (/.f64 t (/.f64 (*.f64 (*.f64 l l) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 6 points increase in error, 4 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2))) (/.f64 t (/.f64 (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (cos.f64 k)) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2))) (/.f64 t (/.f64 (Rewrite<= *-commutative_binary64 (*.f64 (cos.f64 k) (pow.f64 l 2))) (*.f64 (pow.f64 (/.f64 -1 k) -2) (pow.f64 (sin.f64 k) 2))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2))) (/.f64 t (/.f64 (*.f64 (cos.f64 k) (pow.f64 l 2)) (Rewrite<= *-commutative_binary64 (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 (/.f64 -1 k) -2)))))): 0 points increase in error, 0 points decrease in error
      (fma.f64 2 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2))) (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 (/.f64 -1 k) -2))) (*.f64 (cos.f64 k) (pow.f64 l 2))))): 3 points increase in error, 8 points decrease in error
      (Rewrite<= fma-def_binary64 (+.f64 (*.f64 2 (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 t 2))) (*.f64 (cos.f64 k) (pow.f64 l 2)))) (/.f64 (*.f64 t (*.f64 (pow.f64 (sin.f64 k) 2) (pow.f64 (/.f64 -1 k) -2))) (*.f64 (cos.f64 k) (pow.f64 l 2))))): 0 points increase in error, 0 points decrease in error

    if -3.7000000000000002e-56 < k < 2.6000000000000001e-78

    1. Initial program 33.9

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

    2. Simplified47.9

      \[\leadsto \color{blue}{\frac{2}{\frac{{t}^{3}}{\ell \cdot \ell} \cdot \left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \left(\sin k \cdot \tan k\right)\right)}} \]
      Proof
      (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (*.f64 (sin.f64 k) (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (*.f64 (sin.f64 k) (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (*.f64 (sin.f64 k) (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (*.f64 (sin.f64 k) (tan.f64 k))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (Rewrite<= *-commutative_binary64 (*.f64 (*.f64 (sin.f64 k) (tan.f64 k)) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1))))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k))) (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)))): 1 points increase in error, 3 points decrease in error
      (/.f64 2 (*.f64 (Rewrite<= associate-*l*_binary64 (*.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))): 3 points increase in error, 32 points decrease in error

    3. Taylor expanded in k around 0 46.6

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

    4. Simplified46.5

      \[\leadsto \frac{2}{\color{blue}{2 \cdot \frac{t}{\frac{\ell \cdot \ell}{\left(k \cdot k\right) \cdot \left(t \cdot t\right)}}}} \]
      Proof
      (*.f64 2 (/.f64 t (/.f64 (*.f64 l l) (*.f64 (*.f64 k k) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 t (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 (*.f64 k k) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 t (/.f64 (pow.f64 l 2) (*.f64 (Rewrite<= unpow2_binary64 (pow.f64 k 2)) (*.f64 t t))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (/.f64 t (/.f64 (pow.f64 l 2) (*.f64 (pow.f64 k 2) (Rewrite<= unpow2_binary64 (pow.f64 t 2)))))): 0 points increase in error, 0 points decrease in error
      (*.f64 2 (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t (*.f64 (pow.f64 k 2) (pow.f64 t 2))) (pow.f64 l 2)))): 7 points increase in error, 5 points decrease in error

    5. Applied egg-rr5.2

      \[\leadsto \color{blue}{\frac{\frac{\ell}{k \cdot t}}{1} \cdot \frac{\frac{\ell}{k \cdot t}}{t}} \]

    if 2.50000000000000004e121 < k

    1. Initial program 33.3

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

    2. Simplified33.3

      \[\leadsto \color{blue}{\frac{2}{\left(\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{{t}^{3}}{\ell \cdot \ell}\right) \cdot \left(\sin k \cdot \tan k\right)}} \]
      Proof
      (/.f64 2 (*.f64 (*.f64 (+.f64 2 (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (+.f64 (Rewrite<= metadata-eval (+.f64 1 1)) (pow.f64 (/.f64 k t) 2)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= associate-+r+_binary64 (+.f64 1 (+.f64 1 (pow.f64 (/.f64 k t) 2)))) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (*.f64 (*.f64 (Rewrite<= +-commutative_binary64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1)) (/.f64 (pow.f64 t 3) (*.f64 l l))) (*.f64 (sin.f64 k) (tan.f64 k)))): 0 points increase in error, 0 points decrease in error
      (/.f64 2 (Rewrite<= associate-*r*_binary64 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (*.f64 (sin.f64 k) (tan.f64 k)))))): 0 points increase in error, 3 points decrease in error
      (/.f64 2 (*.f64 (+.f64 (+.f64 1 (pow.f64 (/.f64 k t) 2)) 1) (Rewrite<= associate-*l*_binary64 (*.f64 (*.f64 (/.f64 (pow.f64 t 3) (*.f64 l l)) (sin.f64 k)) (tan.f64 k))))): 3 points increase in error, 32 points decrease in error
      (/.f64 2 (Rewrite<= *-commutative_binary64 (*.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 points increase in error, 0 points decrease in error

    3. Taylor expanded in t around 0 22.2

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

    4. Simplified6.1

      \[\leadsto \frac{2}{\color{blue}{\frac{t}{\frac{\ell}{k} \cdot \frac{\ell}{k}}} \cdot \left(\sin k \cdot \tan k\right)} \]
      Proof
      (/.f64 t (*.f64 (/.f64 l k) (/.f64 l k))): 0 points increase in error, 0 points decrease in error
      (/.f64 t (Rewrite<= times-frac_binary64 (/.f64 (*.f64 l l) (*.f64 k k)))): 66 points increase in error, 17 points decrease in error
      (/.f64 t (/.f64 (Rewrite<= unpow2_binary64 (pow.f64 l 2)) (*.f64 k k))): 0 points increase in error, 0 points decrease in error
      (/.f64 t (/.f64 (pow.f64 l 2) (Rewrite<= unpow2_binary64 (pow.f64 k 2)))): 0 points increase in error, 0 points decrease in error
      (Rewrite<= associate-/l*_binary64 (/.f64 (*.f64 t (pow.f64 k 2)) (pow.f64 l 2))): 15 points increase in error, 12 points decrease in error

    5. Applied egg-rr2.8

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

    6. Applied egg-rr2.7

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

  4. Final simplification5.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -5.9 \cdot 10^{+144}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \tan k}\right)\\ \mathbf{elif}\;k \leq -3.7 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{t}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{t} \cdot \frac{\ell}{t}}, \frac{t}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot {\left(\frac{-1}{k}\right)}^{-2}}}\right)}\\ \mathbf{elif}\;k \leq 2.6 \cdot 10^{-78}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{k \cdot t}}{t}\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{+121}:\\ \;\;\;\;\frac{2}{\mathsf{fma}\left(2, \frac{t}{\cos k} \cdot \frac{{\sin k}^{2}}{\frac{\ell}{t} \cdot \frac{\ell}{t}}, \frac{t}{\frac{\cos k \cdot \left(\ell \cdot \ell\right)}{{\sin k}^{2} \cdot {\left(\frac{-1}{k}\right)}^{-2}}}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\frac{\left(t \cdot \sin k\right) \cdot \left(\tan k \cdot \frac{k}{\ell}\right)}{\frac{\ell}{k}}}\\ \end{array} \]

Alternatives

Alternative 1
Error6.3
Cost52548
\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\ell}{\tan k}\\ \mathbf{if}\;t \leq -2.55 \cdot 10^{-110}:\\ \;\;\;\;{\left(\frac{\frac{\sqrt[3]{t_2}}{\sqrt[3]{t_1}} \cdot \sqrt[3]{\ell \cdot 2}}{t \cdot \sqrt[3]{\sin k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 4.1 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+180}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{t_2}{t_1}} \cdot \frac{1}{\frac{t}{\sqrt[3]{\frac{\ell \cdot 2}{\sin k}}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \end{array} \]
Alternative 2
Error6.0
Cost40140
\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}\\ \mathbf{if}\;t \leq -1.1 \cdot 10^{-11}:\\ \;\;\;\;{\left(\frac{t_1}{\frac{t}{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}}\right)}^{3}\\ \mathbf{elif}\;t \leq 1.6 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.15 \cdot 10^{+181}:\\ \;\;\;\;{\left(t_1 \cdot \frac{1}{\frac{t}{\sqrt[3]{\frac{\ell \cdot 2}{\sin k}}}}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \end{array} \]
Alternative 3
Error6.0
Cost40012
\[\begin{array}{l} t_1 := {\left(\sqrt[3]{\frac{\ell \cdot 2}{\sin k}} \cdot \frac{\sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}}{t}\right)}^{3}\\ \mathbf{if}\;t \leq -2 \cdot 10^{-11}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 1.06 \cdot 10^{+15}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 8.2 \cdot 10^{+179}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \end{array} \]
Alternative 4
Error6.0
Cost40012
\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\frac{\ell}{\tan k}}{2 + {\left(\frac{k}{t}\right)}^{2}}}\\ \mathbf{if}\;t \leq -2.4 \cdot 10^{-11}:\\ \;\;\;\;{\left(\frac{t_1}{\frac{t}{\sqrt[3]{\ell \cdot \frac{2}{\sin k}}}}\right)}^{3}\\ \mathbf{elif}\;t \leq 6.5 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 2 \cdot 10^{+179}:\\ \;\;\;\;{\left(\sqrt[3]{\frac{\ell \cdot 2}{\sin k}} \cdot \frac{t_1}{t}\right)}^{3}\\ \mathbf{else}:\\ \;\;\;\;\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \end{array} \]
Alternative 5
Error7.5
Cost27472
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-20}:\\ \;\;\;\;\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\frac{t_2}{\frac{\ell}{\tan k}}}\\ \mathbf{elif}\;t \leq 1.9 \cdot 10^{+17}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\frac{t_2 \cdot \frac{\tan k}{\ell}}{\frac{\ell}{\frac{{t}^{3}}{\frac{2}{\sin k}}}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 6
Error7.5
Cost27080
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ \mathbf{if}\;t \leq -5.6 \cdot 10^{+102}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq -2.2 \cdot 10^{-17}:\\ \;\;\;\;\frac{2}{\sin k \cdot {t}^{3}} \cdot \frac{\ell}{\frac{t_2}{\frac{\ell}{\tan k}}}\\ \mathbf{elif}\;t \leq 3 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+100}:\\ \;\;\;\;\frac{1}{\frac{t_2 \cdot \frac{\tan k}{\ell}}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(\sin k \cdot 0.5\right)\right)}}}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 7
Error6.9
Cost21264
\[\begin{array}{l} t_1 := \frac{1}{\frac{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \frac{\tan k}{\ell}}{\frac{\ell}{\left(t \cdot t\right) \cdot \left(t \cdot \left(\sin k \cdot 0.5\right)\right)}}}\\ t_2 := \frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \mathbf{if}\;t \leq -3.4 \cdot 10^{+106}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq -8 \cdot 10^{-18}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 9 \cdot 10^{+14}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{elif}\;t \leq 1.05 \cdot 10^{+100}:\\ \;\;\;\;t_1\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 8
Error9.6
Cost14288
\[\begin{array}{l} t_1 := 2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k \cdot \left(t \cdot \left(\sin k \cdot \tan k\right)\right)}\right)\\ t_2 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;k \leq -61000000:\\ \;\;\;\;t_1\\ \mathbf{elif}\;k \leq 2.5 \cdot 10^{-53}:\\ \;\;\;\;t_2 \cdot \frac{t_2}{t}\\ \mathbf{elif}\;k \leq 6.5 \cdot 10^{-19}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{elif}\;k \leq 6.1 \cdot 10^{+38}:\\ \;\;\;\;t_2 \cdot \frac{\frac{\ell}{t}}{k \cdot t}\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]
Alternative 9
Error9.2
Cost14024
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.1 \cdot 10^{-11}:\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{elif}\;t \leq 1.28 \cdot 10^{-51}:\\ \;\;\;\;\frac{\ell}{k} \cdot \left(\frac{2}{t} \cdot \frac{\frac{\ell}{k}}{\sin k \cdot \tan k}\right)\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{\sqrt{t}}{t_1}}\right)}^{2}\\ \end{array} \]
Alternative 10
Error8.8
Cost14024
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-11}:\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{+18}:\\ \;\;\;\;\frac{2}{\left(\sin k \cdot \tan k\right) \cdot \left(\frac{k}{\ell} \cdot \left(t \cdot \frac{k}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{\sqrt{t}}{t_1}}\right)}^{2}\\ \end{array} \]
Alternative 11
Error18.6
Cost13704
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_1 \cdot \frac{t_1}{t}\\ \mathbf{elif}\;t \leq 3.5 \cdot 10^{-64}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\frac{\ell}{{k}^{3}}}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;{\left(\frac{1}{\frac{\sqrt{t}}{t_1}}\right)}^{2}\\ \end{array} \]
Alternative 12
Error18.7
Cost7432
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.25 \cdot 10^{-73}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\ell} \cdot {k}^{3}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 13
Error18.6
Cost7432
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{\frac{t}{\frac{\ell}{{k}^{3}}}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 14
Error19.2
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.7 \cdot 10^{-72}:\\ \;\;\;\;2 \cdot \frac{{k}^{-4}}{\frac{\frac{t}{\ell}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 15
Error18.9
Cost7304
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-67}:\\ \;\;\;\;\frac{2}{\frac{t}{\ell} \cdot \frac{{k}^{4}}{\ell}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 16
Error19.6
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.06 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 2.9 \cdot 10^{-56}:\\ \;\;\;\;\frac{2}{\left(k \cdot \left(k \cdot t\right)\right) \cdot \left(\frac{k}{\ell} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 17
Error19.2
Cost1224
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_2 := t_1 \cdot \frac{t_1}{t}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-100}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 5.8 \cdot 10^{-52}:\\ \;\;\;\;\frac{2}{\frac{\frac{k \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;t_2\\ \end{array} \]
Alternative 18
Error35.5
Cost832
\[\frac{\ell}{k \cdot k} \cdot \frac{\frac{\ell}{t}}{t \cdot t} \]
Alternative 19
Error34.6
Cost832
\[\frac{\frac{\ell}{t}}{k \cdot k} \cdot \frac{\ell}{t \cdot t} \]
Alternative 20
Error24.5
Cost832
\[\frac{\ell}{k \cdot t} \cdot \frac{\frac{\ell}{t}}{k \cdot t} \]
Alternative 21
Error24.2
Cost832
\[\begin{array}{l} t_1 := \frac{\ell}{k \cdot t}\\ t_1 \cdot \frac{t_1}{t} \end{array} \]

Error

Reproduce

herbie shell --seed 2022325 
(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))))