Average Error: 48.0 → 13.5

Time: 29.7s

Precision: binary64

Cost: 21122

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

↓

\[\begin{array}{l} \mathbf{if}\;k \leq -1.937105691112254 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.3396666814857636 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

\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}
\mathbf{if}\;k \leq -1.937105691112254 \cdot 10^{-158}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\

\mathbf{elif}\;k \leq 1.3396666814857636 \cdot 10^{-113}:\\
\;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\

\mathbf{else}:\\
\;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\

\end{array}

(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
 (if (<= k -1.937105691112254e-158)
   (* 2.0 (/ (* (cos k) (/ l (/ k l))) (* k (* t (pow (sin k) 2.0)))))
   (if (<= k 1.3396666814857636e-113)
     (/
      2.0
      (* (pow (/ k t) 2.0) (* (tan k) (* (/ t l) (* (sin k) (/ (* t t) l))))))
     (* 2.0 (/ (* (cos k) (/ l (/ k l))) (* k (* t (pow (sin k) 2.0))))))))

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 tmp;
	if (k <= -1.937105691112254e-158) {
		tmp = 2.0 * ((cos(k) * (l / (k / l))) / (k * (t * pow(sin(k), 2.0))));
	} else if (k <= 1.3396666814857636e-113) {
		tmp = 2.0 / (pow((k / t), 2.0) * (tan(k) * ((t / l) * (sin(k) * ((t * t) / l)))));
	} else {
		tmp = 2.0 * ((cos(k) * (l / (k / l))) / (k * (t * pow(sin(k), 2.0))));
	}
	return tmp;
}

Error

The X axis uses an exponential scale — Bits error versus `t`

Try it out

In
Out

Enter valid numbers for all inputs

Alternatives

Alternative 1
Error	40.1
Cost	99136

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

Alternative 2
Error	22.9
Cost	79744

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

Alternative 3
Error	37.7
Cost	72064

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

Alternative 4
Error	22.9
Cost	66176

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

Alternative 5
Error	51.5
Cost	66112

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

Alternative 6
Error	22.9
Cost	65664

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

Alternative 7
Error	21.0
Cost	65664

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

Alternative 8
Error	50.1
Cost	65536

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

Alternative 9
Error	50.8
Cost	65536

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

Alternative 10
Error	21.0
Cost	65408

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

Alternative 11
Error	22.9
Cost	65408

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

Alternative 12
Error	28.4
Cost	53184

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

Alternative 13
Error	27.3
Cost	53184

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

Alternative 14
Error	41.6
Cost	52736

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

Alternative 15
Error	43.1
Cost	46400

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

Alternative 16
Error	50.4
Cost	46208

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

Alternative 17
Error	43.1
Cost	46144

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

Alternative 18
Error	18.6
Cost	46016

\[2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot \left(\left(t \cdot {\left(\sqrt[3]{\sin k}\right)}^{4}\right) \cdot {\left(\sqrt[3]{\sin k}\right)}^{2}\right)}\]

Alternative 19
Error	18.6
Cost	40192

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

Alternative 20
Error	37.6
Cost	40000

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

Alternative 21
Error	36.1
Cost	39744

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

Alternative 22
Error	43.2
Cost	39744

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

Alternative 23
Error	47.7
Cost	39744

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

Alternative 24
Error	34.1
Cost	39744

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

Alternative 25
Error	13.4
Cost	39680

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

Alternative 26
Error	42.7
Cost	39552

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

Alternative 27
Error	46.4
Cost	39296

\[2 \cdot e^{2 \cdot \log \left(\frac{\ell}{k}\right) + \log \left(\frac{\cos k}{t \cdot {\sin k}^{2}}\right)}\]

Alternative 28
Error	47.9
Cost	33408

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

Alternative 29
Error	50.3
Cost	33344

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

Alternative 30
Error	47.9
Cost	33280

\[\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \left(\sin k \cdot \left(\frac{{t}^{1.5}}{\ell} \cdot \frac{{t}^{1.5}}{\ell}\right)\right)\right)}\]

Alternative 31
Error	40.9
Cost	33216

\[2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot \left(\left(\sin k \cdot \sqrt{t}\right) \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}\]

Alternative 32
Error	43.1
Cost	33216

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(\left(\sin k \cdot \sqrt{t}\right) \cdot \left(\sin k \cdot \sqrt{t}\right)\right)}\]

Alternative 33
Error	52.8
Cost	33216

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

Alternative 34
Error	43.3
Cost	33152

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

Alternative 35
Error	43.1
Cost	33152

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

Alternative 36
Error	40.0
Cost	33152

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

Alternative 37
Error	40.2
Cost	33152

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

Alternative 38
Error	38.9
Cost	33152

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

Alternative 39
Error	30.8
Cost	33088

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

Alternative 40
Error	30.3
Cost	33088

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

Alternative 41
Error	29.0
Cost	33088

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

Alternative 42
Error	29.5
Cost	33088

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

Alternative 43
Error	27.5
Cost	33088

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

Alternative 44
Error	41.7
Cost	33024

\[2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot e^{\log \left(t \cdot {\sin k}^{2}\right)}}\]

Alternative 45
Error	29.5
Cost	33024

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot \log \left(e^{{\sin k}^{2}}\right)\right)}\]

Alternative 46
Error	48.0
Cost	26944

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

Alternative 47
Error	46.5
Cost	26944

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

Alternative 48
Error	37.4
Cost	26816

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

Alternative 49
Error	38.0
Cost	26816

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

Alternative 50
Error	38.0
Cost	26816

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

Alternative 51
Error	40.0
Cost	26688

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

Alternative 52
Error	40.2
Cost	26688

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

Alternative 53
Error	40.2
Cost	26688

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

Alternative 54
Error	38.0
Cost	26688

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

Alternative 55
Error	23.7
Cost	26624

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot \sqrt[3]{{\sin k}^{6}}\right)\right)}\]

Alternative 56
Error	25.6
Cost	26624

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot \sqrt[3]{{\sin k}^{6}}\right)}\]

Alternative 57
Error	34.0
Cost	20480

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

Alternative 58
Error	34.0
Cost	20480

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

Alternative 59
Error	35.0
Cost	20480

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

Alternative 60
Error	36.0
Cost	20480

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

Alternative 61
Error	34.5
Cost	20480

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

Alternative 62
Error	36.3
Cost	20480

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

Alternative 63
Error	18.5
Cost	20352

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

Alternative 64
Error	18.4
Cost	20352

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

Alternative 65
Error	22.9
Cost	20352

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

Alternative 66
Error	18.2
Cost	20288

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

Alternative 67
Error	22.7
Cost	20288

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

Alternative 68
Error	41.6
Cost	20288

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

Alternative 69
Error	22.7
Cost	20224

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

Alternative 70
Error	13.2
Cost	20224

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

Alternative 71
Error	20.8
Cost	20224

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

Alternative 72
Error	13.2
Cost	20224

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

Alternative 73
Error	20.8
Cost	20224

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

Alternative 74
Error	21.6
Cost	20224

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

Alternative 75
Error	21.5
Cost	20224

\[2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k \cdot k}}{t \cdot {\sin k}^{2}}\]

Alternative 76
Error	22.7
Cost	20224

\[2 \cdot \frac{\ell \cdot \ell}{\frac{k \cdot k}{\frac{\cos k}{t \cdot {\sin k}^{2}}}}\]

Alternative 77
Error	22.7
Cost	20224

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

Alternative 78
Error	22.8
Cost	20224

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

Alternative 79
Error	18.4
Cost	20224

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

Alternative 80
Error	18.4
Cost	20224

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

Alternative 81
Error	18.4
Cost	20224

\[2 \cdot \frac{\frac{\ell \cdot \ell}{k}}{\frac{k}{\frac{\cos k}{t \cdot {\sin k}^{2}}}}\]

Alternative 82
Error	30.5
Cost	7488

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot \left(t \cdot \left(k \cdot k\right)\right)\right)}\]

Alternative 83
Error	29.7
Cost	7488

\[2 \cdot \frac{\cos k \cdot \frac{\ell \cdot \ell}{k}}{k \cdot \left(t \cdot \left(k \cdot k\right)\right)}\]

Alternative 84
Error	30.5
Cost	7488

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot \left(k \cdot k\right)\right)}\]

Alternative 85
Error	30.5
Cost	7488

\[2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k \cdot \left(k \cdot \left(k \cdot \left(k \cdot t\right)\right)\right)}\]

Alternative 86
Error	29.8
Cost	7424

\[\frac{\ell}{\frac{t}{\ell}} \cdot \left(\frac{2}{{k}^{4}} - \frac{0.3333333333333333}{k \cdot k}\right)\]

Alternative 87
Error	32.0
Cost	7040

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

Alternative 88
Error	30.3
Cost	7040

\[\frac{2}{\frac{{k}^{4}}{\frac{\ell}{\frac{t}{\ell}}}}\]

Alternative 89
Error	61.9
Cost	64

\[1\]

Alternative 90
Error	34.4
Cost	64

\[0\]

Alternative 91
Error	61.9
Cost	64

\[-1\]

Error

Derivation

Split input into 2 regimes
if k < -1.937105691112254e-158 or 1.3396666814857636e-113 < k
1. Initial program 47.1
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Simplified38.7
  \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}}\]
3. Taylor expanded around 0 20.7
  \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot {\ell}^{2}}{{k}^{2} \cdot \left(t \cdot {\sin k}^{2}\right)}}\]
4. Simplified20.7
  \[\leadsto \color{blue}{2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\left(k \cdot k\right) \cdot \left(t \cdot {\sin k}^{2}\right)}}\]
5. Using strategy rm
6. Applied associate-*l*_binary64_34518.7
  \[\leadsto 2 \cdot \frac{\left(\ell \cdot \ell\right) \cdot \cos k}{\color{blue}{k \cdot \left(k \cdot \left(t \cdot {\sin k}^{2}\right)\right)}}\]
7. Using strategy rm
8. Applied associate-/r*_binary64_34816.4
  \[\leadsto 2 \cdot \color{blue}{\frac{\frac{\left(\ell \cdot \ell\right) \cdot \cos k}{k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}\]
9. Simplified16.4
  \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell \cdot \ell}{k} \cdot \cos k}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\]
10. Using strategy rm
11. Applied associate-/l*_binary64_34911.0
  \[\leadsto 2 \cdot \frac{\color{blue}{\frac{\ell}{\frac{k}{\ell}}} \cdot \cos k}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\]
12. Simplified11.0
  \[\leadsto \color{blue}{2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}}\]
if -1.937105691112254e-158 < k < 1.3396666814857636e-113
1. Initial program 64.0
  \[\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot \left(\left(1 + {\left(\frac{k}{t}\right)}^{2}\right) - 1\right)}\]
2. Simplified63.5
  \[\leadsto \color{blue}{\frac{2}{\left(\left(\frac{{t}^{3}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}}\]
3. Using strategy rm
4. Applied cube-mult_binary64_43463.5
  \[\leadsto \frac{2}{\left(\left(\frac{\color{blue}{t \cdot \left(t \cdot t\right)}}{\ell \cdot \ell} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\]
5. Applied times-frac_binary64_41059.5
  \[\leadsto \frac{2}{\left(\left(\color{blue}{\left(\frac{t}{\ell} \cdot \frac{t \cdot t}{\ell}\right)} \cdot \sin k\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\]
6. Applied associate-*l*_binary64_34557.3
  \[\leadsto \frac{2}{\left(\color{blue}{\left(\frac{t}{\ell} \cdot \left(\frac{t \cdot t}{\ell} \cdot \sin k\right)\right)} \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\]
7. Simplified57.3
  \[\leadsto \frac{2}{\left(\left(\frac{t}{\ell} \cdot \color{blue}{\left(\sin k \cdot \frac{t \cdot t}{\ell}\right)}\right) \cdot \tan k\right) \cdot {\left(\frac{k}{t}\right)}^{2}}\]
8. Simplified57.3
  \[\leadsto \color{blue}{\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}}\]
Recombined 2 regimes into one program.
Final simplification13.5
\[\leadsto \begin{array}{l} \mathbf{if}\;k \leq -1.937105691112254 \cdot 10^{-158}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \mathbf{elif}\;k \leq 1.3396666814857636 \cdot 10^{-113}:\\ \;\;\;\;\frac{2}{{\left(\frac{k}{t}\right)}^{2} \cdot \left(\tan k \cdot \left(\frac{t}{\ell} \cdot \left(\sin k \cdot \frac{t \cdot t}{\ell}\right)\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;2 \cdot \frac{\cos k \cdot \frac{\ell}{\frac{k}{\ell}}}{k \cdot \left(t \cdot {\sin k}^{2}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2021042 
(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))))

Error

Try it out

Alternatives

Error

Derivation

`if k < -1.937105691112254e-158 or 1.3396666814857636e-113 < k`

`if -1.937105691112254e-158 < k < 1.3396666814857636e-113`

Reproduce