Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 10.0

Time: 12.9s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\sin k}\\ t_2 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_3 := \ell \cdot \sqrt{2}\\ t_4 := \frac{t_3}{t \cdot \left(t_1 \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{t_2}\right)\right)}\\ t_5 := \sqrt[3]{\tan k \cdot t_2}\\ t_6 := \frac{t_4}{t \cdot t_5}\\ \mathbf{if}\;t \leq -3 \cdot 10^{-73}:\\ \;\;\;\;t_6 \cdot \frac{t_4}{t_1}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-207}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;t_6 \cdot \frac{\frac{t_3}{t \cdot \left(t_1 \cdot t_5\right)}}{t_1}\\ \end{array} \]

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 (cbrt (sin k)))
        (t_2 (+ 2.0 (pow (/ k t) 2.0)))
        (t_3 (* l (sqrt 2.0)))
        (t_4 (/ t_3 (* t (* t_1 (* (cbrt (tan k)) (cbrt t_2))))))
        (t_5 (cbrt (* (tan k) t_2)))
        (t_6 (/ t_4 (* t t_5))))
   (if (<= t -3e-73)
     (* t_6 (/ t_4 t_1))
     (if (<= t 5.6e-207)
       (*
        2.0
        (/ (* (cos k) (pow l 2.0)) (* (pow k 2.0) (* t (pow (sin k) 2.0)))))
       (* t_6 (/ (/ t_3 (* t (* t_1 t_5))) t_1))))))

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 = cbrt(sin(k));
	double t_2 = 2.0 + pow((k / t), 2.0);
	double t_3 = l * sqrt(2.0);
	double t_4 = t_3 / (t * (t_1 * (cbrt(tan(k)) * cbrt(t_2))));
	double t_5 = cbrt((tan(k) * t_2));
	double t_6 = t_4 / (t * t_5);
	double tmp;
	if (t <= -3e-73) {
		tmp = t_6 * (t_4 / t_1);
	} else if (t <= 5.6e-207) {
		tmp = 2.0 * ((cos(k) * pow(l, 2.0)) / (pow(k, 2.0) * (t * pow(sin(k), 2.0))));
	} else {
		tmp = t_6 * ((t_3 / (t * (t_1 * t_5))) / t_1);
	}
	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.cbrt(Math.sin(k));
	double t_2 = 2.0 + Math.pow((k / t), 2.0);
	double t_3 = l * Math.sqrt(2.0);
	double t_4 = t_3 / (t * (t_1 * (Math.cbrt(Math.tan(k)) * Math.cbrt(t_2))));
	double t_5 = Math.cbrt((Math.tan(k) * t_2));
	double t_6 = t_4 / (t * t_5);
	double tmp;
	if (t <= -3e-73) {
		tmp = t_6 * (t_4 / t_1);
	} else if (t <= 5.6e-207) {
		tmp = 2.0 * ((Math.cos(k) * Math.pow(l, 2.0)) / (Math.pow(k, 2.0) * (t * Math.pow(Math.sin(k), 2.0))));
	} else {
		tmp = t_6 * ((t_3 / (t * (t_1 * t_5))) / t_1);
	}
	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 = cbrt(sin(k))
	t_2 = Float64(2.0 + (Float64(k / t) ^ 2.0))
	t_3 = Float64(l * sqrt(2.0))
	t_4 = Float64(t_3 / Float64(t * Float64(t_1 * Float64(cbrt(tan(k)) * cbrt(t_2)))))
	t_5 = cbrt(Float64(tan(k) * t_2))
	t_6 = Float64(t_4 / Float64(t * t_5))
	tmp = 0.0
	if (t <= -3e-73)
		tmp = Float64(t_6 * Float64(t_4 / t_1));
	elseif (t <= 5.6e-207)
		tmp = Float64(2.0 * Float64(Float64(cos(k) * (l ^ 2.0)) / Float64((k ^ 2.0) * Float64(t * (sin(k) ^ 2.0)))));
	else
		tmp = Float64(t_6 * Float64(Float64(t_3 / Float64(t * Float64(t_1 * t_5))) / t_1));
	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_] := Block[{t$95$1 = N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$2 = N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$3 = N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$4 = N[(t$95$3 / N[(t * N[(t$95$1 * N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[t$95$2, 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, Block[{t$95$5 = N[Power[N[(N[Tan[k], $MachinePrecision] * t$95$2), $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$6 = N[(t$95$4 / N[(t * t$95$5), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -3e-73], N[(t$95$6 * N[(t$95$4 / t$95$1), $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-207], N[(2.0 * N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / N[(N[Power[k, 2.0], $MachinePrecision] * N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(t$95$6 * N[(N[(t$95$3 / N[(t * N[(t$95$1 * t$95$5), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / t$95$1), $MachinePrecision]), $MachinePrecision]]]]]]]]]

Derivation

1. Split input into 3 regimes

2. if t < -3e-73

   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 22.9

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

   3. Applied egg-rr 17.1

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

   4. Applied egg-rr 3.6

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

   5. Applied egg-rr 3.6

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

   6. Applied egg-rr 3.6

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

   if -3e-73 < t < 5.59999999999999986e-207

   1. Initial program 60.3

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

   2. Simplified 60.1

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

   3. Taylor expanded in k around inf 28.4

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

   if 5.59999999999999986e-207 < t

   1. Initial program 29.2

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

   2. Simplified 28.9

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

   3. Applied egg-rr 20.4

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

   4. Applied egg-rr 7.9

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

   5. Applied egg-rr 7.9

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

4. Final simplification 10.0

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -3 \cdot 10^{-73}:\\ \;\;\;\;\frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)}}{t \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)}}{\sqrt[3]{\sin k}}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-207}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \left(\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}\right)\right)}}{t \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}} \cdot \frac{\frac{\ell \cdot \sqrt{2}}{t \cdot \left(\sqrt[3]{\sin k} \cdot \sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}\right)}}{\sqrt[3]{\sin k}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022209 
(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))))