Toniolo and Linder, Equation (10-)

Average Error: 47.4 → 0.7

Time: 45.8s

Precision: binary64

Cost: 33289

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-56} \lor \neg \left(k \leq 6.6 \cdot 10^{-17}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right) \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

FPCore

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

↓

(FPCore (t l k)
 :precision binary64
 (if (or (<= k -2e-56) (not (<= k 6.6e-17)))
   (*
    2.0
    (/
     (* (/ (cos k) k) l)
     (/ (* (expm1 (log1p (pow (sin k) 2.0))) t) (/ l k))))
   (fma
    2.0
    (/ (/ (/ l k) (- k)) (* t (* k (/ (- k) l))))
    (* (* -2.0 (* (/ l k) (/ l k))) (* (/ t t) (/ 0.16666666666666666 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 ((k <= -2e-56) || !(k <= 6.6e-17)) {
		tmp = 2.0 * (((cos(k) / k) * l) / ((expm1(log1p(pow(sin(k), 2.0))) * t) / (l / k)));
	} else {
		tmp = fma(2.0, (((l / k) / -k) / (t * (k * (-k / l)))), ((-2.0 * ((l / k) * (l / k))) * ((t / t) * (0.16666666666666666 / 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 ((k <= -2e-56) || !(k <= 6.6e-17))
		tmp = Float64(2.0 * Float64(Float64(Float64(cos(k) / k) * l) / Float64(Float64(expm1(log1p((sin(k) ^ 2.0))) * t) / Float64(l / k))));
	else
		tmp = fma(2.0, Float64(Float64(Float64(l / k) / Float64(-k)) / Float64(t * Float64(k * Float64(Float64(-k) / l)))), Float64(Float64(-2.0 * Float64(Float64(l / k) * Float64(l / k))) * Float64(Float64(t / t) * Float64(0.16666666666666666 / t))));
	end
	return 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[Or[LessEqual[k, -2e-56], N[Not[LessEqual[k, 6.6e-17]], $MachinePrecision]], N[(2.0 * N[(N[(N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision] * l), $MachinePrecision] / N[(N[(N[(Exp[N[Log[1 + N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]], $MachinePrecision]] - 1), $MachinePrecision] * t), $MachinePrecision] / N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 * N[(N[(N[(l / k), $MachinePrecision] / (-k)), $MachinePrecision] / N[(t * N[(k * N[((-k) / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] + N[(N[(-2.0 * N[(N[(l / k), $MachinePrecision] * N[(l / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(t / t), $MachinePrecision] * N[(0.16666666666666666 / t), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]

Derivation

1. Split input into 2 regimes

2. if k < -2.0000000000000001e-56 or 6.60000000000000001e-17 < k

   1. Initial program 44.4

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

   2. Simplified 34.4

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

      [Start] 44.4	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 44.5	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 44.5	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 44.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 43.1	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 43.1	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 43.1	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 34.4	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 34.4	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 34.4	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around inf 18.9

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

   4. Simplified 19.2

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

      [Start] 18.9	\[ 2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left({\sin k}^{2} \cdot t\right)} \]
      times-frac [=>] 19.2	\[ 2 \cdot \color{blue}{\left(\frac{\cos k}{{k}^{2}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right)} \]
      unpow2 [=>] 19.2	\[ 2 \cdot \left(\frac{\cos k}{\color{blue}{k \cdot k}} \cdot \frac{{\ell}^{2}}{{\sin k}^{2} \cdot t}\right) \]
      unpow2 [=>] 19.2	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\color{blue}{\ell \cdot \ell}}{{\sin k}^{2} \cdot t}\right) \]
      *-commutative [=>] 19.2	\[ 2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \frac{\ell \cdot \ell}{\color{blue}{t \cdot {\sin k}^{2}}}\right) \]

   5. Applied egg-rr 3.8

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

   6. Applied egg-rr 0.6

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

   7. Applied egg-rr 0.6

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

   if -2.0000000000000001e-56 < k < 6.60000000000000001e-17

   1. Initial program 62.6

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

   2. Simplified 50.2

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

      [Start] 62.6	\[ \frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)} \]
      associate-/r* [=>] 62.6	\[ \color{blue}{\frac{\frac{2}{\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1}} \]
      *-commutative [=>] 62.6	\[ \frac{\frac{2}{\color{blue}{\tan k \cdot \left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*l/ [=>] 62.8	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\frac{{t}^{3} \cdot \sin k}{\ell \cdot \ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      times-frac [=>] 61.5	\[ \frac{\frac{2}{\tan k \cdot \color{blue}{\left(\frac{{t}^{3}}{\ell} \cdot \frac{\sin k}{\ell}\right)}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      associate-*r* [=>] 61.4	\[ \frac{\frac{2}{\color{blue}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}}{\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1} \]
      +-commutative [=>] 61.4	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{\left({\left(\frac{k}{t}\right)}^{2} + 1\right)} - 1} \]
      associate--l+ [=>] 50.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2} + \left(1 - 1\right)}} \]
      metadata-eval [=>] 50.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{{\left(\frac{k}{t}\right)}^{2} + \color{blue}{0}} \]
      +-rgt-identity [=>] 50.2	\[ \frac{\frac{2}{\left(\tan k \cdot \frac{{t}^{3}}{\ell}\right) \cdot \frac{\sin k}{\ell}}}{\color{blue}{{\left(\frac{k}{t}\right)}^{2}}} \]

   3. Taylor expanded in k around 0 49.9

      \[\leadsto \color{blue}{-2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   4. Simplified 41.0

      \[\leadsto \color{blue}{\mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)} \]
      Proof

      [Start] 49.9	\[ -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}} + 2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} \]
      +-commutative [=>] 49.9	\[ \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t} + -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}} \]
      fma-def [=>] 49.9	\[ \color{blue}{\mathsf{fma}\left(2, \frac{{\ell}^{2}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right)} \]
      unpow2 [=>] 49.9	\[ \mathsf{fma}\left(2, \frac{\color{blue}{\ell \cdot \ell}}{{k}^{4} \cdot t}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]
      *-commutative [=>] 49.9	\[ \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{\color{blue}{t \cdot {k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]
      times-frac [=>] 48.0	\[ \mathsf{fma}\left(2, \color{blue}{\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}}, -2 \cdot \frac{{\ell}^{2} \cdot \left(-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t\right)}{{k}^{2} \cdot {t}^{2}}\right) \]
      times-frac [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, -2 \cdot \color{blue}{\left(\frac{{\ell}^{2}}{{k}^{2}} \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right)}\right) \]
      associate-*r* [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \color{blue}{\left(-2 \cdot \frac{{\ell}^{2}}{{k}^{2}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}}\right) \]
      unpow2 [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \frac{\color{blue}{\ell \cdot \ell}}{{k}^{2}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]
      unpow2 [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \frac{\ell \cdot \ell}{\color{blue}{k \cdot k}}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]
      times-frac [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \color{blue}{\left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)}\right) \cdot \frac{-0.16666666666666666 \cdot t + 0.3333333333333333 \cdot t}{{t}^{2}}\right) \]
      distribute-rgt-out [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \frac{\color{blue}{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}}{{t}^{2}}\right) \]
      unpow2 [=>] 45.4	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \frac{t \cdot \left(-0.16666666666666666 + 0.3333333333333333\right)}{\color{blue}{t \cdot t}}\right) \]
      times-frac [=>] 41.0	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \color{blue}{\left(\frac{t}{t} \cdot \frac{-0.16666666666666666 + 0.3333333333333333}{t}\right)}\right) \]
      metadata-eval [=>] 41.0	\[ \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{\color{blue}{0.16666666666666666}}{t}\right)\right) \]

   5. Applied egg-rr 29.0

      \[\leadsto \mathsf{fma}\left(2, \frac{\ell}{t} \cdot \color{blue}{\left(\frac{1}{k \cdot k} \cdot \frac{\ell}{k \cdot k}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]

   6. Applied egg-rr 1.5

      \[\leadsto \mathsf{fma}\left(2, \color{blue}{\frac{\frac{\frac{\ell}{k}}{-k}}{\left(\frac{k}{\ell} \cdot k\right) \cdot \left(-t\right)}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]
3. Recombined 2 regimes into one program.

4. Final simplification 0.7

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -2 \cdot 10^{-56} \lor \neg \left(k \leq 6.6 \cdot 10^{-17}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{\mathsf{expm1}\left(\mathsf{log1p}\left({\sin k}^{2}\right)\right) \cdot t}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternatives

Alternative 1
Error	0.5
Cost	20489

\[\begin{array}{l} \mathbf{if}\;k \leq -7.5 \cdot 10^{-56} \lor \neg \left(k \leq 5 \cdot 10^{-28}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{t \cdot \left({\sin k}^{2} \cdot \frac{k}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 2
Error	0.6
Cost	20488

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \frac{\cos k}{k} \cdot \ell\\ \mathbf{if}\;k \leq -4 \cdot 10^{-55}:\\ \;\;\;\;2 \cdot \frac{t_2}{t \cdot \left(t_1 \cdot \frac{k}{\ell}\right)}\\ \mathbf{elif}\;k \leq 2 \cdot 10^{-20}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{\frac{t_1 \cdot t}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 3
Error	4.3
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -32000000000 \lor \neg \left(k \leq 0.0005\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{k \cdot \left(0.5 \cdot \left(t \cdot \frac{1 - \cos \left(k + k\right)}{\ell}\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 4
Error	4.3
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -32000000000 \lor \neg \left(k \leq 0.00125\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\left(0.5 - \frac{\cos \left(k + k\right)}{2}\right) \cdot \left(k \cdot \frac{t}{\ell}\right)}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 5
Error	1.4
Cost	14409

\[\begin{array}{l} \mathbf{if}\;k \leq -32000000000 \lor \neg \left(k \leq 0.00095\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{t \cdot \left(0.5 - \frac{\cos \left(k + k\right)}{2}\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \end{array} \]

Alternative 6
Error	4.3
Cost	14408

\[\begin{array}{l} t_1 := \cos \left(k + k\right)\\ t_2 := \frac{\cos k}{k} \cdot \ell\\ \mathbf{if}\;k \leq -32000000000:\\ \;\;\;\;2 \cdot \frac{t_2}{k \cdot \left(0.5 \cdot \left(t \cdot \frac{1 - t_1}{\ell}\right)\right)}\\ \mathbf{elif}\;k \leq 0.0014:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{t_2}{k \cdot \left(\left(0.5 - \frac{t_1}{2}\right) \cdot \frac{t}{\ell}\right)}\\ \end{array} \]

Alternative 7
Error	20.6
Cost	14084

\[\begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 10^{+34}:\\ \;\;\;\;\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right)\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\ell \cdot \frac{1}{k}}{\frac{{\sin k}^{2} \cdot t}{\frac{\ell}{k}}}\\ \end{array} \]

Alternative 8
Error	21.5
Cost	8512

\[\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{-k}}{t \cdot \left(k \cdot \frac{-k}{\ell}\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]

Alternative 9
Error	22.6
Cost	8384

\[\mathsf{fma}\left(2, \frac{\frac{\ell}{k} \cdot \frac{\frac{\ell}{k}}{k \cdot t}}{k}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]

Alternative 10
Error	22.3
Cost	8384

\[\mathsf{fma}\left(2, \frac{\frac{\ell}{k}}{\left(k \cdot \frac{k}{\ell}\right) \cdot \left(k \cdot t\right)}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]

Alternative 11
Error	22.0
Cost	8384

\[\mathsf{fma}\left(2, \frac{\frac{\frac{\ell}{k}}{k \cdot t}}{k \cdot \frac{k}{\ell}}, \left(-2 \cdot \left(\frac{\ell}{k} \cdot \frac{\ell}{k}\right)\right) \cdot \left(\frac{t}{t} \cdot \frac{0.16666666666666666}{t}\right)\right) \]

Alternative 12
Error	22.4
Cost	8265

\[\begin{array}{l} \mathbf{if}\;t \leq -4.4 \cdot 10^{-38} \lor \neg \left(t \leq 4.3 \cdot 10^{+55}\right):\\ \;\;\;\;2 \cdot \frac{\frac{\cos k}{k} \cdot \ell}{\frac{k \cdot \left(k \cdot t\right)}{\frac{\ell}{k}}}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \left(\frac{\cos k}{k \cdot k} \cdot \left(\frac{\ell}{\frac{t}{\ell} \cdot \left(k \cdot k\right)} + 0.3333333333333333 \cdot \left(\ell \cdot \frac{\ell}{t}\right)\right)\right)\\ \end{array} \]

Alternative 13
Error	23.7
Cost	7488

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

Alternative 14
Error	24.5
Cost	7296

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

Alternative 15
Error	25.9
Cost	960

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

Alternative 16
Error	25.3
Cost	960

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

Reproduce

herbie shell --seed 2023125 
(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))))