Toniolo and Linder, Equation (10-)

Average Error: 48.2 → 16.1

Time: 25.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{k}}\\ \mathbf{if}\;\ell \cdot \ell \leq 4.9736400097324085 \cdot 10^{+305}:\\ \;\;\;\;2 \cdot \frac{t_1 \cdot t_1}{\frac{k}{\frac{\sqrt[3]{\cos k \cdot \frac{\ell \cdot \ell}{k}}}{t \cdot {\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

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 (/ (* (cos k) (pow l 2.0)) k))))
   (if (<= (* l l) 4.9736400097324085e+305)
     (*
      2.0
      (/
       (* t_1 t_1)
       (/ k (/ (cbrt (* (cos k) (/ (* l l) k))) (* t (pow (sin k) 2.0))))))
     (/
      2.0
      (*
       (* (* (/ t l) (* (sin k) (/ (* t t) l))) (tan k))
       (pow (/ k t) 2.0))))))

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 = cbrt(((cos(k) * pow(l, 2.0)) / k));
	double tmp;
	if ((l * l) <= 4.9736400097324085e+305) {
		tmp = 2.0 * ((t_1 * t_1) / (k / (cbrt((cos(k) * ((l * l) / k))) / (t * pow(sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((((t / l) * (sin(k) * ((t * t) / l))) * tan(k)) * pow((k / t), 2.0));
	}
	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 = Math.cbrt(((Math.cos(k) * Math.pow(l, 2.0)) / k));
	double tmp;
	if ((l * l) <= 4.9736400097324085e+305) {
		tmp = 2.0 * ((t_1 * t_1) / (k / (Math.cbrt((Math.cos(k) * ((l * l) / k))) / (t * Math.pow(Math.sin(k), 2.0)))));
	} else {
		tmp = 2.0 / ((((t / l) * (Math.sin(k) * ((t * t) / l))) * Math.tan(k)) * Math.pow((k / t), 2.0));
	}
	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 = cbrt(Float64(Float64(cos(k) * (l ^ 2.0)) / k))
	tmp = 0.0
	if (Float64(l * l) <= 4.9736400097324085e+305)
		tmp = Float64(2.0 * Float64(Float64(t_1 * t_1) / Float64(k / Float64(cbrt(Float64(cos(k) * Float64(Float64(l * l) / k))) / Float64(t * (sin(k) ^ 2.0))))));
	else
		tmp = Float64(2.0 / Float64(Float64(Float64(Float64(t / l) * Float64(sin(k) * Float64(Float64(t * t) / l))) * tan(k)) * (Float64(k / t) ^ 2.0)));
	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[Power[N[(N[(N[Cos[k], $MachinePrecision] * N[Power[l, 2.0], $MachinePrecision]), $MachinePrecision] / k), $MachinePrecision], 1/3], $MachinePrecision]}, If[LessEqual[N[(l * l), $MachinePrecision], 4.9736400097324085e+305], N[(2.0 * N[(N[(t$95$1 * t$95$1), $MachinePrecision] / N[(k / N[(N[Power[N[(N[Cos[k], $MachinePrecision] * N[(N[(l * l), $MachinePrecision] / k), $MachinePrecision]), $MachinePrecision], 1/3], $MachinePrecision] / N[(t * N[Power[N[Sin[k], $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(2.0 / N[(N[(N[(N[(t / l), $MachinePrecision] * N[(N[Sin[k], $MachinePrecision] * N[(N[(t * t), $MachinePrecision] / l), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[Tan[k], $MachinePrecision]), $MachinePrecision] * N[Power[N[(k / t), $MachinePrecision], 2.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 2 regimes

2. if (*.f64 l l) < 4.9736400097324085e305

   1. Initial program 45.4

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

   2. Simplified 35.8

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

   3. Taylor expanded in t around 0 15.5

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

   4. Applied unpow2_binary64 15.5

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

   5. Applied associate-*l*_binary64 12.9

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

   6. Applied associate-/r*_binary64 10.2

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

   7. Applied add-cube-cbrt_binary64 10.5

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

   8. Applied associate-/l*_binary64 10.5

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

   9. Simplified 9.9

      \[\leadsto 2 \cdot \frac{\sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{k}} \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{k}}}{\color{blue}{\frac{k}{\frac{\sqrt[3]{\frac{\ell \cdot \ell}{k} \cdot \cos k}}{t \cdot {\sin k}^{2}}}}} \]

   if 4.9736400097324085e305 < (*.f64 l l)

   1. Initial program 63.9

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

   2. Simplified 63.9

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

   3. Applied cube-mult_binary64 63.9

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

   4. Applied times-frac_binary64 50.2

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

   5. Applied associate-*l*_binary64 50.2

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

4. Final simplification 16.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;\ell \cdot \ell \leq 4.9736400097324085 \cdot 10^{+305}:\\ \;\;\;\;2 \cdot \frac{\sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{k}} \cdot \sqrt[3]{\frac{\cos k \cdot {\ell}^{2}}{k}}}{\frac{k}{\frac{\sqrt[3]{\cos k \cdot \frac{\ell \cdot \ell}{k}}}{t \cdot {\sin k}^{2}}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{2}{\left(\left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022138 
(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))))