Average Error: 42.9 → 9.2
Time: 21.6s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 1.673826578300523 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x}{x - 1} + 2 \cdot \frac{1}{x - 1}}}\\ \end{array}\]
\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 \leq -1.2351728058409963 \cdot 10^{-41}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\

\mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\

\mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\

\mathbf{elif}\;t \leq 1.673826578300523 \cdot 10^{-305}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\

\mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\

\mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x}{x - 1} + 2 \cdot \frac{1}{x - 1}}}\\

\end{array}
(FPCore (x l t)
 :precision binary64
 (/
  (* (sqrt 2.0) t)
  (sqrt (- (* (/ (+ x 1.0) (- x 1.0)) (+ (* l l) (* 2.0 (* t t)))) (* l l)))))
↓
(FPCore (x l t)
 :precision binary64
 (if (<= t -1.2351728058409963e-41)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t -1.5844530436624705e-154)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+ (* 2.0 (/ l (/ x l))) (+ (* 2.0 (* t t)) (* 4.0 (/ (* t t) x))))))
     (if (<= t -4.222777320075483e-236)
       (/
        (* t (sqrt 2.0))
        (-
         (- (* t (sqrt (+ 2.0 (/ 4.0 x)))))
         (* (sqrt (/ 1.0 (+ 2.0 (/ 4.0 x)))) (/ (* l l) (* t x)))))
       (if (<= t 1.673826578300523e-305)
         (/ (* t (sqrt 2.0)) (* (sqrt (/ 1.0 x)) (* (sqrt 2.0) l)))
         (if (<= t 3.385594735519463e-144)
           (/
            (* t (sqrt 2.0))
            (+
             (* t (sqrt 2.0))
             (+
              (* 2.0 (/ t (* (sqrt 2.0) x)))
              (/ (pow l 2.0) (* t (* (sqrt 2.0) x))))))
           (if (<= t 2.2674113060888642e+126)
             (/
              (* t (sqrt 2.0))
              (sqrt
               (+
                (+ (* 2.0 (* t t)) (* 4.0 (/ (* t t) x)))
                (* 2.0 (* l (/ l x))))))
             (/
              (* t (sqrt 2.0))
              (*
               t
               (sqrt
                (+ (* 2.0 (/ x (- x 1.0))) (* 2.0 (/ 1.0 (- x 1.0))))))))))))))
double code(double x, double l, double t) {
	return (sqrt(2.0) * t) / sqrt((((x + 1.0) / (x - 1.0)) * ((l * l) + (2.0 * (t * t)))) - (l * l));
}
↓
double code(double x, double l, double t) {
	double tmp;
	if (t <= -1.2351728058409963e-41) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= -1.5844530436624705e-154) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (l / (x / l))) + ((2.0 * (t * t)) + (4.0 * ((t * t) / x))));
	} else if (t <= -4.222777320075483e-236) {
		tmp = (t * sqrt(2.0)) / (-(t * sqrt(2.0 + (4.0 / x))) - (sqrt(1.0 / (2.0 + (4.0 / x))) * ((l * l) / (t * x))));
	} else if (t <= 1.673826578300523e-305) {
		tmp = (t * sqrt(2.0)) / (sqrt(1.0 / x) * (sqrt(2.0) * l));
	} else if (t <= 3.385594735519463e-144) {
		tmp = (t * sqrt(2.0)) / ((t * sqrt(2.0)) + ((2.0 * (t / (sqrt(2.0) * x))) + (pow(l, 2.0) / (t * (sqrt(2.0) * x)))));
	} else if (t <= 2.2674113060888642e+126) {
		tmp = (t * sqrt(2.0)) / sqrt(((2.0 * (t * t)) + (4.0 * ((t * t) / x))) + (2.0 * (l * (l / x))));
	} else {
		tmp = (t * sqrt(2.0)) / (t * sqrt((2.0 * (x / (x - 1.0))) + (2.0 * (1.0 / (x - 1.0)))));
	}
	return tmp;
}

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 7 regimes

  2. if t < -1.2351728058409963e-41

    1. Initial program 39.8

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

    2. Taylor expanded around -inf 6.3

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

    3. Simplified6.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]

    if -1.2351728058409963e-41 < t < -1.58445304366247051e-154

    1. Initial program 32.6

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

    2. Taylor expanded around inf 10.0

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

    3. Simplified10.0

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

    5. Applied associate-/l*_binary64_234.3

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

    if -1.58445304366247051e-154 < t < -4.2227773200754832e-236

    1. Initial program 61.6

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

    2. Taylor expanded around inf 36.4

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

    3. Simplified36.4

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

    4. Taylor expanded around -inf 24.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{-\left(\sqrt{4 \cdot \frac{1}{x} + 2} \cdot t + \sqrt{\frac{1}{4 \cdot \frac{1}{x} + 2}} \cdot \frac{{\ell}^{2}}{x \cdot t}\right)}}\]

    5. Simplified24.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}}\]

    if -4.2227773200754832e-236 < t < 1.67382657830052298e-305

    1. Initial program 62.9

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

    2. Taylor expanded around inf 29.8

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

    3. Simplified29.8

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

    4. Taylor expanded around inf 32.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{\sqrt{\frac{1}{x}} \cdot \left(\ell \cdot \sqrt{2}\right)}}\]

    if 1.67382657830052298e-305 < t < 3.38559473551946292e-144

    1. Initial program 60.4

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

    2. Taylor expanded around inf 24.1

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

    if 3.38559473551946292e-144 < t < 2.26741130608886417e126

    1. Initial program 24.5

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

    2. Taylor expanded around inf 10.7

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

    3. Simplified10.7

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

    5. Applied *-un-lft-identity_binary64_7810.7

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \frac{\ell \cdot \ell}{\color{blue}{1 \cdot x}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\]

    6. Applied times-frac_binary64_845.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot \color{blue}{\left(\frac{\ell}{1} \cdot \frac{\ell}{x}\right)} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\]

    7. Simplified5.1

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

    if 2.26741130608886417e126 < t

    1. Initial program 55.1

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

    2. Taylor expanded around inf 2.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
  3. Recombined 7 regimes into one program.

  4. Final simplification9.2

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.2351728058409963 \cdot 10^{-41}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -1.5844530436624705 \cdot 10^{-154}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq -4.222777320075483 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(-t \cdot \sqrt{2 + \frac{4}{x}}\right) - \sqrt{\frac{1}{2 + \frac{4}{x}}} \cdot \frac{\ell \cdot \ell}{t \cdot x}}\\ \mathbf{elif}\;t \leq 1.673826578300523 \cdot 10^{-305}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\frac{1}{x}} \cdot \left(\sqrt{2} \cdot \ell\right)}\\ \mathbf{elif}\;t \leq 3.385594735519463 \cdot 10^{-144}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 2.2674113060888642 \cdot 10^{+126}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right) + 2 \cdot \left(\ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x}{x - 1} + 2 \cdot \frac{1}{x - 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021044 
(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)))))