Toniolo and Linder, Equation (10+)

Average Error: 32.3 → 9.4

Time: 12.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{4}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{if}\;t \leq -1 \cdot 10^{-52}:\\ \;\;\;\;t_1\\ \mathbf{elif}\;t \leq 10^{-50}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;t_1\\ \end{array} \]

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
         (*
          l
          (pow
           (*
            (cbrt l)
            (/
             (/ (/ (sqrt (cbrt 4.0)) t) (cbrt (sin k)))
             (* (cbrt (tan k)) (cbrt (+ 2.0 (pow (/ k t) 2.0))))))
           3.0))))
   (if (<= t -1e-52)
     t_1
     (if (<= t 1e-50)
       (* l (* 2.0 (/ (/ (* (/ l k) (/ (cos k) k)) t) (pow (sin k) 2.0))))
       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 = l * pow((cbrt(l) * (((sqrt(cbrt(4.0)) / t) / cbrt(sin(k))) / (cbrt(tan(k)) * cbrt((2.0 + pow((k / t), 2.0)))))), 3.0);
	double tmp;
	if (t <= -1e-52) {
		tmp = t_1;
	} else if (t <= 1e-50) {
		tmp = l * (2.0 * ((((l / k) * (cos(k) / k)) / t) / pow(sin(k), 2.0)));
	} else {
		tmp = 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 = l * Math.pow((Math.cbrt(l) * (((Math.sqrt(Math.cbrt(4.0)) / t) / Math.cbrt(Math.sin(k))) / (Math.cbrt(Math.tan(k)) * Math.cbrt((2.0 + Math.pow((k / t), 2.0)))))), 3.0);
	double tmp;
	if (t <= -1e-52) {
		tmp = t_1;
	} else if (t <= 1e-50) {
		tmp = l * (2.0 * ((((l / k) * (Math.cos(k) / k)) / t) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = 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 = Float64(l * (Float64(cbrt(l) * Float64(Float64(Float64(sqrt(cbrt(4.0)) / t) / cbrt(sin(k))) / Float64(cbrt(tan(k)) * cbrt(Float64(2.0 + (Float64(k / t) ^ 2.0)))))) ^ 3.0))
	tmp = 0.0
	if (t <= -1e-52)
		tmp = t_1;
	elseif (t <= 1e-50)
		tmp = Float64(l * Float64(2.0 * Float64(Float64(Float64(Float64(l / k) * Float64(cos(k) / k)) / t) / (sin(k) ^ 2.0))));
	else
		tmp = 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[(l * N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(N[(N[(N[Sqrt[N[Power[4.0, 1/3], $MachinePrecision]], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision] / N[(N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision] * N[Power[N[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -1e-52], t$95$1, If[LessEqual[t, 1e-50], N[(l * N[(2.0 * N[(N[(N[(N[(l / k), $MachinePrecision] * N[(N[Cos[k], $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], t$95$1]]]

Derivation

1. Split input into 2 regimes

2. if t < -1e-52 or 1.00000000000000001e-50 < t

   1. Initial program 22.3

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

   2. Simplified 18.5

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

   3. Applied egg-rr 14.2

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

   4. Applied egg-rr 7.7

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

   5. Applied egg-rr 7.6

      \[\leadsto \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\color{blue}{\sqrt{\sqrt[3]{4}}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\tan k \cdot \left(2 + {\left(\frac{k}{t}\right)}^{2}\right)}}\right)}^{3} \]

   6. Applied egg-rr 7.6

      \[\leadsto \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{4}}}{t}}{\sqrt[3]{\sin k}}}{\color{blue}{\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}}\right)}^{3} \]

   if -1e-52 < t < 1.00000000000000001e-50

   1. Initial program 55.7

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

   2. Simplified 55.5

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

   3. Taylor expanded in t around 0 22.5

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

   4. Simplified 13.6

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

4. Final simplification 9.4

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1 \cdot 10^{-52}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{4}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \mathbf{elif}\;t \leq 10^{-50}:\\ \;\;\;\;\ell \cdot \left(2 \cdot \frac{\frac{\frac{\ell}{k} \cdot \frac{\cos k}{k}}{t}}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\sqrt{\sqrt[3]{4}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\tan k} \cdot \sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}}}\right)}^{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022211 
(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))))