Average Error: 43.1 → 9.3
Time: 7.8s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -1.08948086886903181 \cdot 10^{153}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 2.79370913846099375 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le 8.03190726921288569 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.62286774086380161 \cdot 10^{118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]
\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}
↓
\begin{array}{l}
\mathbf{if}\;t \le -1.08948086886903181 \cdot 10^{153}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 2.79370913846099375 \cdot 10^{-206}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\

\mathbf{elif}\;t \le 8.03190726921288569 \cdot 10^{-169}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\mathbf{elif}\;t \le 1.62286774086380161 \cdot 10^{118}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}}}\\

\mathbf{else}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\

\end{array}
double code(double x, double l, double t) {
	return ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))) * ((double) (((double) (l * l)) + ((double) (2.0 * ((double) (t * t)))))))) - ((double) (l * l))))))));
}
↓
double code(double x, double l, double t) {
	double VAR;
	if ((t <= -1.0894808688690318e+153)) {
		VAR = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0)))))))) - ((double) (((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))) + ((double) (t * ((double) sqrt(2.0))))))))));
	} else {
		double VAR_1;
		if ((t <= 2.793709138460994e-206)) {
			VAR_1 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) pow(t, 2.0)) + ((double) (((double) (l / ((double) (((double) cbrt(x)) * ((double) cbrt(x)))))) * ((double) (l / ((double) cbrt(x))))))))))))))));
		} else {
			double VAR_2;
			if ((t <= 8.031907269212886e-169)) {
				VAR_2 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))) + ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))))) + ((double) (((double) (((double) sqrt(2.0)) * t)) - ((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0))))))))))))));
			} else {
				double VAR_3;
				if ((t <= 1.6228677408638016e+118)) {
					VAR_3 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) sqrt(((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) pow(t, 2.0)) + ((double) (l / ((double) (x / l)))))))))))))) * ((double) sqrt(((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) pow(t, 2.0)) + ((double) (l / ((double) (x / l))))))))))))))))));
				} else {
					VAR_3 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))) + ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))))) + ((double) (((double) (((double) sqrt(2.0)) * t)) - ((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0))))))))))))));
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

Derivation

  1. Split input into 4 regimes

  2. if t < -1.08948086886903181e153

    1. Initial program 62.3

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 62.8

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

    3. Simplified62.8

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

    4. Taylor expanded around -inf 1.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}}\]

    if -1.08948086886903181e153 < t < 2.79370913846099375e-206

    1. Initial program 39.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 19.4

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

    3. Simplified19.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
    4. Using strategy rm

    5. Applied add-cube-cbrt19.4

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

    6. Applied add-sqr-sqrt42.2

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

    7. Applied unpow-prod-down42.2

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

    8. Applied times-frac40.4

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

    9. Simplified40.4

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

    10. Simplified15.5

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

    if 2.79370913846099375e-206 < t < 8.03190726921288569e-169 or 1.62286774086380161e118 < t

    1. Initial program 55.2

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(2 \cdot \frac{t}{\sqrt{2} \cdot {x}^{2}} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)\right) - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}}}\]

    3. Simplified5.4

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}}\]

    if 8.03190726921288569e-169 < t < 1.62286774086380161e118

    1. Initial program 27.0

      \[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]

    2. Taylor expanded around inf 11.1

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

    3. Simplified11.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{{\ell}^{2}}{x}\right)}}}\]
    4. Using strategy rm

    5. Applied unpow211.1

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

    6. Applied associate-/l*6.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)}}\]
    7. Using strategy rm

    8. Applied add-sqr-sqrt6.2

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

    9. Applied sqrt-prod6.3

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -1.08948086886903181 \cdot 10^{153}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 2.79370913846099375 \cdot 10^{-206}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right)}}\\ \mathbf{elif}\;t \le 8.03190726921288569 \cdot 10^{-169}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 1.62286774086380161 \cdot 10^{118}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}} \cdot \sqrt{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \frac{\ell}{\frac{x}{\ell}}\right)}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020155 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2.0) t) (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))