Average Error: 43.0 → 8.6
Time: 14.8s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \le -3.794989872061021 \cdot 10^{143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \frac{t}{x \cdot \sqrt{2}}\right) - \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.3289760264210936 \cdot 10^{110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{x \cdot \sqrt{2}} + \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}\right)}\\ \end{array}\]
\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 -3.794989872061021 \cdot 10^{143}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \frac{t}{x \cdot \sqrt{2}}\right) - \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + t \cdot \sqrt{2}\right)}\\

\mathbf{elif}\;t \le 3.3289760264210936 \cdot 10^{110}:\\
\;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\

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

\end{array}
double code(double x, double l, double t) {
	return (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((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 <= -3.794989872061021e+143)) {
		VAR = (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) ((t / ((double) (((double) pow(x, 2.0)) * ((double) pow(((double) sqrt(2.0)), 3.0))))) - (t / ((double) (x * ((double) sqrt(2.0))))))))) - ((double) (((double) (2.0 * (t / ((double) (((double) pow(x, 2.0)) * ((double) sqrt(2.0))))))) + ((double) (t * ((double) sqrt(2.0)))))))));
	} else {
		double VAR_1;
		if ((t <= 3.3289760264210936e+110)) {
			VAR_1 = (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (2.0 * ((double) ((l / ((double) (((double) cbrt(x)) * ((double) cbrt(x))))) * (l / ((double) cbrt(x))))))) + ((double) (((double) (4.0 * (((double) pow(t, 2.0)) / x))) + ((double) (2.0 * ((double) pow(t, 2.0)))))))))));
		} else {
			VAR_1 = (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) ((t / ((double) (x * ((double) sqrt(2.0))))) + (t / ((double) (((double) pow(x, 2.0)) * ((double) sqrt(2.0))))))))) + ((double) (((double) (((double) sqrt(2.0)) * t)) - ((double) (2.0 * (t / ((double) (((double) pow(x, 2.0)) * ((double) pow(((double) sqrt(2.0)), 3.0))))))))))));
		}
		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 3 regimes

  2. if t < -3.794989872061021e143

    1. Initial program 59.2

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

    2. Taylor expanded around -inf 2.0

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

    3. Simplified2.0

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

    if -3.794989872061021e143 < t < 3.3289760264210936e110

    1. Initial program 35.8

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

    2. Taylor expanded around inf 16.7

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

    4. Applied add-cube-cbrt16.8

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

    5. Applied add-sqr-sqrt40.3

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

    6. Applied unpow-prod-down40.3

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

    7. Applied times-frac38.0

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

    8. Simplified38.0

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

    9. Simplified12.2

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

    if 3.3289760264210936e110 < t

    1. Initial program 52.2

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

    2. Taylor expanded around inf 2.9

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

    3. Simplified2.9

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

  4. Final simplification8.6

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -3.794989872061021 \cdot 10^{143}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}} - \frac{t}{x \cdot \sqrt{2}}\right) - \left(2 \cdot \frac{t}{{x}^{2} \cdot \sqrt{2}} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le 3.3289760264210936 \cdot 10^{110}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{x \cdot \sqrt{2}} + \frac{t}{{x}^{2} \cdot \sqrt{2}}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{x}^{2} \cdot {\left(\sqrt{2}\right)}^{3}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020182 
(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)))))