Toniolo and Linder, Equation (10+)

Average Error: 32.4 → 9.2

Time: 12.3s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := 2 + {\left(\frac{k}{t}\right)}^{2}\\ t_2 := \frac{\frac{{2}^{0.3333333333333333}}{t}}{\sqrt[3]{\sin k}}\\ \mathbf{if}\;t \leq -2.45 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{t_2}{\sqrt[3]{t_1} \cdot \sqrt[3]{\tan k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-61}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{t_2}{\sqrt[3]{t_1 \cdot \tan k}}\right)}^{3}\\ \end{array} \]

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 (+ 2.0 (pow (/ k t) 2.0)))
        (t_2 (/ (/ (pow 2.0 0.3333333333333333) t) (cbrt (sin k)))))
   (if (<= t -2.45e-55)
     (* l (pow (* (cbrt l) (/ t_2 (* (cbrt t_1) (cbrt (tan k))))) 3.0))
     (if (<= t 5.6e-61)
       (* l (* (/ 2.0 (* k (* t k))) (/ (* l (cos k)) (pow (sin k) 2.0))))
       (* l (pow (* (cbrt l) (/ t_2 (cbrt (* t_1 (tan k))))) 3.0))))))

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 = 2.0 + pow((k / t), 2.0);
	double t_2 = (pow(2.0, 0.3333333333333333) / t) / cbrt(sin(k));
	double tmp;
	if (t <= -2.45e-55) {
		tmp = l * pow((cbrt(l) * (t_2 / (cbrt(t_1) * cbrt(tan(k))))), 3.0);
	} else if (t <= 5.6e-61) {
		tmp = l * ((2.0 / (k * (t * k))) * ((l * cos(k)) / pow(sin(k), 2.0)));
	} else {
		tmp = l * pow((cbrt(l) * (t_2 / cbrt((t_1 * tan(k))))), 3.0);
	}
	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 = 2.0 + Math.pow((k / t), 2.0);
	double t_2 = (Math.pow(2.0, 0.3333333333333333) / t) / Math.cbrt(Math.sin(k));
	double tmp;
	if (t <= -2.45e-55) {
		tmp = l * Math.pow((Math.cbrt(l) * (t_2 / (Math.cbrt(t_1) * Math.cbrt(Math.tan(k))))), 3.0);
	} else if (t <= 5.6e-61) {
		tmp = l * ((2.0 / (k * (t * k))) * ((l * Math.cos(k)) / Math.pow(Math.sin(k), 2.0)));
	} else {
		tmp = l * Math.pow((Math.cbrt(l) * (t_2 / Math.cbrt((t_1 * Math.tan(k))))), 3.0);
	}
	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(2.0 + (Float64(k / t) ^ 2.0))
	t_2 = Float64(Float64((2.0 ^ 0.3333333333333333) / t) / cbrt(sin(k)))
	tmp = 0.0
	if (t <= -2.45e-55)
		tmp = Float64(l * (Float64(cbrt(l) * Float64(t_2 / Float64(cbrt(t_1) * cbrt(tan(k))))) ^ 3.0));
	elseif (t <= 5.6e-61)
		tmp = Float64(l * Float64(Float64(2.0 / Float64(k * Float64(t * k))) * Float64(Float64(l * cos(k)) / (sin(k) ^ 2.0))));
	else
		tmp = Float64(l * (Float64(cbrt(l) * Float64(t_2 / cbrt(Float64(t_1 * tan(k))))) ^ 3.0));
	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[(2.0 + N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]}, Block[{t$95$2 = N[(N[(N[Power[2.0, 0.3333333333333333], $MachinePrecision] / t), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]}, If[LessEqual[t, -2.45e-55], N[(l * N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(t$95$2 / N[(N[Power[t$95$1, 1/3], $MachinePrecision] * N[Power[N[Tan[k], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision], If[LessEqual[t, 5.6e-61], N[(l * N[(N[(2.0 / N[(k * N[(t * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision] / N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(l * N[Power[N[(N[Power[l, 1/3], $MachinePrecision] * N[(t$95$2 / N[Power[N[(t$95$1 * N[Tan[k], $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision], 3.0], $MachinePrecision]), $MachinePrecision]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 3 regimes

2. if t < -2.45000000000000018e-55

   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 18.1

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

   3. Applied egg-rr 13.8

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

   4. Applied egg-rr 7.6

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

   5. Applied egg-rr 7.4

      \[\leadsto \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\color{blue}{{2}^{0.3333333333333333}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}}\right)}^{3} \]

   6. Applied egg-rr 7.4

      \[\leadsto \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{{2}^{0.3333333333333333}}{t}}{\sqrt[3]{\sin k}}}{\color{blue}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\tan k}}}\right)}^{3} \]

   if -2.45000000000000018e-55 < t < 5.6000000000000002e-61

   1. Initial program 55.9

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

   2. Simplified 56.3

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

   3. Taylor expanded in t around 0 22.4

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

   4. Simplified 12.4

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

   if 5.6000000000000002e-61 < t

   1. Initial program 22.9

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

   2. Simplified 18.9

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

   3. Applied egg-rr 15.1

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

   4. Applied egg-rr 8.5

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

   5. Applied egg-rr 8.3

      \[\leadsto \ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{\color{blue}{{2}^{0.3333333333333333}}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}}\right)}^{3} \]
3. Recombined 3 regimes into one program.

4. Final simplification 9.2

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.45 \cdot 10^{-55}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{{2}^{0.3333333333333333}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{2 + {\left(\frac{k}{t}\right)}^{2}} \cdot \sqrt[3]{\tan k}}\right)}^{3}\\ \mathbf{elif}\;t \leq 5.6 \cdot 10^{-61}:\\ \;\;\;\;\ell \cdot \left(\frac{2}{k \cdot \left(t \cdot k\right)} \cdot \frac{\ell \cdot \cos k}{{\sin k}^{2}}\right)\\ \mathbf{else}:\\ \;\;\;\;\ell \cdot {\left(\sqrt[3]{\ell} \cdot \frac{\frac{\frac{{2}^{0.3333333333333333}}{t}}{\sqrt[3]{\sin k}}}{\sqrt[3]{\left(2 + {\left(\frac{k}{t}\right)}^{2}\right) \cdot \tan k}}\right)}^{3}\\ \end{array} \]

Reproduce

herbie shell --seed 2022162 
(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))))