Average Error: 42.8 → 8.3
Time: 17.3s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -7.925051725780633 \cdot 10^{+105}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.9657881463761506 \cdot 10^{-237}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.0817817285694105 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 7.465776635895058 \cdot 10^{+28}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\ \end{array}\]
\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 -7.925051725780633 \cdot 10^{+105}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\

\mathbf{elif}\;t \leq 2.9657881463761506 \cdot 10^{-237}:\\
\;\;\;\;\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 1.0817817285694105 \cdot 10^{-160}:\\
\;\;\;\;\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 7.465776635895058 \cdot 10^{+28}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + 2 \cdot {t}^{2}}}\\

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

\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 -7.925051725780633e+105)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (* 2.0 (/ 1.0 (+ -1.0 x))) (* 2.0 (/ x (+ -1.0 x))))))))
   (if (<= t 2.9657881463761506e-237)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+ (* 2.0 (/ l (/ x l))) (+ (* 2.0 (* t t)) (* 4.0 (/ (* t t) x))))))
     (if (<= t 1.0817817285694105e-160)
       (/
        (* 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 7.465776635895058e+28)
         (/
          (* t (sqrt 2.0))
          (sqrt
           (+
            (* 2.0 (* (/ l (* (cbrt x) (cbrt x))) (/ l (cbrt x))))
            (* 2.0 (pow t 2.0)))))
         (/
          (* t (sqrt 2.0))
          (*
           t
           (sqrt
            (+ (* 2.0 (/ 1.0 (+ -1.0 x))) (* 2.0 (/ x (+ -1.0 x))))))))))))
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 <= -7.925051725780633e+105) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 * (1.0 / (-1.0 + x))) + (2.0 * (x / (-1.0 + x)))));
	} else if (t <= 2.9657881463761506e-237) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (l / (x / l))) + ((2.0 * (t * t)) + (4.0 * ((t * t) / x))));
	} else if (t <= 1.0817817285694105e-160) {
		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 <= 7.465776635895058e+28) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * ((l / (cbrt(x) * cbrt(x))) * (l / cbrt(x)))) + (2.0 * pow(t, 2.0)));
	} else {
		tmp = (t * sqrt(2.0)) / (t * sqrt((2.0 * (1.0 / (-1.0 + x))) + (2.0 * (x / (-1.0 + x)))));
	}
	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 5 regimes

  2. if t < -7.92505172578063283e105

    1. Initial program 51.3

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

    2. Taylor expanded around -inf 2.6

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

    if -7.92505172578063283e105 < t < 2.9657881463761506e-237

    1. Initial program 40.1

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

    2. Taylor expanded around inf 17.8

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

    3. Simplified17.8

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

    5. Applied associate-/l*_binary64_2314.2

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

    if 2.9657881463761506e-237 < t < 1.08178172856941047e-160

    1. Initial program 62.6

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

    2. Taylor expanded around inf 23.3

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

    if 1.08178172856941047e-160 < t < 7.46577663589505797e28

    1. Initial program 30.2

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

    2. Taylor expanded around inf 9.1

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

    3. Simplified9.1

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

    5. Applied add-cube-cbrt_binary64_1139.1

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

    6. Applied times-frac_binary64_844.6

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

    7. Taylor expanded around inf 5.0

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

    if 7.46577663589505797e28 < t

    1. Initial program 42.9

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

    2. Taylor expanded around inf 4.1

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
  3. Recombined 5 regimes into one program.

  4. Final simplification8.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -7.925051725780633 \cdot 10^{+105}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\ \mathbf{elif}\;t \leq 2.9657881463761506 \cdot 10^{-237}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{\ell}{\frac{x}{\ell}} + \left(2 \cdot \left(t \cdot t\right) + 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq 1.0817817285694105 \cdot 10^{-160}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2} + \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + \frac{{\ell}^{2}}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 7.465776635895058 \cdot 10^{+28}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \left(\frac{\ell}{\sqrt[3]{x} \cdot \sqrt[3]{x}} \cdot \frac{\ell}{\sqrt[3]{x}}\right) + 2 \cdot {t}^{2}}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021019 
(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)))))