Toniolo and Linder, Equation (10-)

Average Error: 47.5 → 8.5

Time: 24.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := {\sin k}^{2}\\ t_2 := \sqrt[3]{\cos k}\\ t_3 := \left(\ell \cdot \sqrt{2}\right) \cdot \frac{t_2}{\sqrt{k}}\\ \mathbf{if}\;k \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \sqrt[3]{{\cos k}^{2}}\right)}{t_1 \cdot \frac{t}{\frac{t_2}{k}}}\\ \mathbf{elif}\;k \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq -7.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\frac{t_1}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t_3}{t_1} \cdot \frac{t_3}{t \cdot \frac{k}{t_2}}\\ \end{array} \]

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 (pow (sin k) 2.0))
        (t_2 (cbrt (cos k)))
        (t_3 (* (* l (sqrt 2.0)) (/ t_2 (sqrt k)))))
   (if (<= k -1.55e-5)
     (/
      (* (* l (/ l k)) (* 2.0 (cbrt (pow (cos k) 2.0))))
      (* t_1 (/ t (/ t_2 k))))
     (if (<= k -1.75e-65)
       (* 2.0 (* (/ l t) (/ l (pow k 4.0))))
       (if (<= k -7.2e-149)
         (* (/ 2.0 (* t (* k k))) (/ l (/ t_1 (* l (cos k)))))
         (* (/ t_3 t_1) (/ t_3 (* t (/ k t_2)))))))))

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 = pow(sin(k), 2.0);
	double t_2 = cbrt(cos(k));
	double t_3 = (l * sqrt(2.0)) * (t_2 / sqrt(k));
	double tmp;
	if (k <= -1.55e-5) {
		tmp = ((l * (l / k)) * (2.0 * cbrt(pow(cos(k), 2.0)))) / (t_1 * (t / (t_2 / k)));
	} else if (k <= -1.75e-65) {
		tmp = 2.0 * ((l / t) * (l / pow(k, 4.0)));
	} else if (k <= -7.2e-149) {
		tmp = (2.0 / (t * (k * k))) * (l / (t_1 / (l * cos(k))));
	} else {
		tmp = (t_3 / t_1) * (t_3 / (t * (k / t_2)));
	}
	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.pow(Math.sin(k), 2.0);
	double t_2 = Math.cbrt(Math.cos(k));
	double t_3 = (l * Math.sqrt(2.0)) * (t_2 / Math.sqrt(k));
	double tmp;
	if (k <= -1.55e-5) {
		tmp = ((l * (l / k)) * (2.0 * Math.cbrt(Math.pow(Math.cos(k), 2.0)))) / (t_1 * (t / (t_2 / k)));
	} else if (k <= -1.75e-65) {
		tmp = 2.0 * ((l / t) * (l / Math.pow(k, 4.0)));
	} else if (k <= -7.2e-149) {
		tmp = (2.0 / (t * (k * k))) * (l / (t_1 / (l * Math.cos(k))));
	} else {
		tmp = (t_3 / t_1) * (t_3 / (t * (k / t_2)));
	}
	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 = sin(k) ^ 2.0
	t_2 = cbrt(cos(k))
	t_3 = Float64(Float64(l * sqrt(2.0)) * Float64(t_2 / sqrt(k)))
	tmp = 0.0
	if (k <= -1.55e-5)
		tmp = Float64(Float64(Float64(l * Float64(l / k)) * Float64(2.0 * cbrt((cos(k) ^ 2.0)))) / Float64(t_1 * Float64(t / Float64(t_2 / k))));
	elseif (k <= -1.75e-65)
		tmp = Float64(2.0 * Float64(Float64(l / t) * Float64(l / (k ^ 4.0))));
	elseif (k <= -7.2e-149)
		tmp = Float64(Float64(2.0 / Float64(t * Float64(k * k))) * Float64(l / Float64(t_1 / Float64(l * cos(k)))));
	else
		tmp = Float64(Float64(t_3 / t_1) * Float64(t_3 / Float64(t * Float64(k / t_2))));
	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], 2.0], $MachinePrecision]}, Block[{t$95$2 = N[Power[N[Cos[k], $MachinePrecision], 1/3], $MachinePrecision]}, Block[{t$95$3 = N[(N[(l * N[Sqrt[2.0], $MachinePrecision]), $MachinePrecision] * N[(t$95$2 / N[Sqrt[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]}, If[LessEqual[k, -1.55e-5], N[(N[(N[(l * N[(l / k), $MachinePrecision]), $MachinePrecision] * N[(2.0 * N[Power[N[Power[N[Cos[k], $MachinePrecision], 2.0], $MachinePrecision], 1/3], $MachinePrecision]), $MachinePrecision]), $MachinePrecision] / N[(t$95$1 * N[(t / N[(t$95$2 / k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -1.75e-65], N[(2.0 * N[(N[(l / t), $MachinePrecision] * N[(l / N[Power[k, 4.0], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], If[LessEqual[k, -7.2e-149], N[(N[(2.0 / N[(t * N[(k * k), $MachinePrecision]), $MachinePrecision]), $MachinePrecision] * N[(l / N[(t$95$1 / N[(l * N[Cos[k], $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision], N[(N[(t$95$3 / t$95$1), $MachinePrecision] * N[(t$95$3 / N[(t * N[(k / t$95$2), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]), $MachinePrecision]]]]]]]

Error

Bits error versus t

Bits error versus l

Bits error versus k

Derivation

1. Split input into 4 regimes

2. if k < -1.55000000000000007e-5

   1. Initial program 43.4

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

   2. Simplified 35.8

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

   3. Taylor expanded in t around 0 19.1

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

   4. Simplified 16.3

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

   5. Applied egg-rr 16.2

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

   6. Applied egg-rr 14.7

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

   7. Taylor expanded in k around inf 14.6

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

   8. Simplified 8.7

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

   if -1.55000000000000007e-5 < k < -1.75000000000000002e-65

   1. Initial program 57.8

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

   2. Simplified 43.7

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

   3. Taylor expanded in t around 0 20.6

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

   4. Simplified 20.7

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

   5. Taylor expanded in k around 0 20.7

      \[\leadsto \color{blue}{2 \cdot \frac{{\ell}^{2}}{{k}^{4} \cdot t}} \]

   6. Simplified 1.4

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

   if -1.75000000000000002e-65 < k < -7.2000000000000004e-149

   1. Initial program 63.3

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

   2. Simplified 61.5

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

   3. Taylor expanded in t around 0 42.9

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

   4. Simplified 43.3

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

   5. Applied egg-rr 43.3

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

   6. Taylor expanded in k around inf 42.9

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

   7. Simplified 21.4

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

   if -7.2000000000000004e-149 < k

   1. Initial program 48.4

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

   2. Simplified 41.1

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

   3. Taylor expanded in t around 0 23.8

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

   4. Simplified 22.0

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

   5. Applied egg-rr 21.8

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

   6. Applied egg-rr 19.8

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

   7. Applied egg-rr 7.9

      \[\leadsto \color{blue}{\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt[3]{\cos k}}{\sqrt{k}}}{{\sin k}^{2}} \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt[3]{\cos k}}{\sqrt{k}}}{t \cdot \frac{k}{\sqrt[3]{\cos k}}}} \]
3. Recombined 4 regimes into one program.

4. Final simplification 8.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.55 \cdot 10^{-5}:\\ \;\;\;\;\frac{\left(\ell \cdot \frac{\ell}{k}\right) \cdot \left(2 \cdot \sqrt[3]{{\cos k}^{2}}\right)}{{\sin k}^{2} \cdot \frac{t}{\frac{\sqrt[3]{\cos k}}{k}}}\\ \mathbf{elif}\;k \leq -1.75 \cdot 10^{-65}:\\ \;\;\;\;2 \cdot \left(\frac{\ell}{t} \cdot \frac{\ell}{{k}^{4}}\right)\\ \mathbf{elif}\;k \leq -7.2 \cdot 10^{-149}:\\ \;\;\;\;\frac{2}{t \cdot \left(k \cdot k\right)} \cdot \frac{\ell}{\frac{{\sin k}^{2}}{\ell \cdot \cos k}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt[3]{\cos k}}{\sqrt{k}}}{{\sin k}^{2}} \cdot \frac{\left(\ell \cdot \sqrt{2}\right) \cdot \frac{\sqrt[3]{\cos k}}{\sqrt{k}}}{t \cdot \frac{k}{\sqrt[3]{\cos k}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022166 
(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))))