Average Error: 42.9 → 12.0
Time: 23.2s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -1.43838900620848 \cdot 10^{-27}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -3.702260431372761 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{elif}\;t \leq 5.463357865450902 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.2822754990186077 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x}{x - 1} + 2 \cdot \frac{1}{x - 1}}}\\ \end{array}\]
\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 -1.43838900620848 \cdot 10^{-27}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\

\mathbf{elif}\;t \leq -3.702260431372761 \cdot 10^{-236}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\

\mathbf{elif}\;t \leq 5.463357865450902 \cdot 10^{-189}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\

\mathbf{elif}\;t \leq 1.2822754990186077 \cdot 10^{+34}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\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 -1.43838900620848e-27)
   (/
    (* t (sqrt 2.0))
    (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
   (if (<= t -3.702260431372761e-236)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+
        (* 2.0 (/ (pow l 2.0) x))
        (+ (* 4.0 (/ (pow t 2.0) x)) (* 2.0 (pow t 2.0))))))
     (if (<= t 5.463357865450902e-189)
       (/ (* t (sqrt 2.0)) (* l (sqrt (+ (/ 2.0 (* x x)) (/ 2.0 x)))))
       (if (<= t 1.2822754990186077e+34)
         (/
          (* t (sqrt 2.0))
          (sqrt
           (+
            (* 2.0 (/ (pow l 2.0) x))
            (+ (* 4.0 (/ (pow t 2.0) x)) (* 2.0 (pow t 2.0))))))
         (/
          (* 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 <= -1.43838900620848e-27) {
		tmp = (t * sqrt(2.0)) / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else if (t <= -3.702260431372761e-236) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (pow(l, 2.0) / x)) + ((4.0 * (pow(t, 2.0) / x)) + (2.0 * pow(t, 2.0))));
	} else if (t <= 5.463357865450902e-189) {
		tmp = (t * sqrt(2.0)) / (l * sqrt((2.0 / (x * x)) + (2.0 / x)));
	} else if (t <= 1.2822754990186077e+34) {
		tmp = (t * sqrt(2.0)) / sqrt((2.0 * (pow(l, 2.0) / x)) + ((4.0 * (pow(t, 2.0) / x)) + (2.0 * pow(t, 2.0))));
	} 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 4 regimes

  2. if t < -1.43838900620848004e-27

    1. Initial program 40.5

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

    2. Taylor expanded around -inf 5.8

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

    3. Simplified5.8

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

    if -1.43838900620848004e-27 < t < -3.70226043137276091e-236 or 5.4633578654509018e-189 < t < 1.28227549901860767e34

    1. Initial program 37.2

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

    2. Taylor expanded around inf 15.6

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

    if -3.70226043137276091e-236 < t < 5.4633578654509018e-189

    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 36.9

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

    3. Simplified36.9

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

    4. Taylor expanded around inf 35.8

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

    5. Simplified35.8

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

    if 1.28227549901860767e34 < t

    1. Initial program 43.6

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

    2. Taylor expanded around inf 4.0

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 \cdot \frac{1}{x - 1} + 2 \cdot \frac{x}{x - 1}}}}\]
  3. Recombined 4 regimes into one program.

  4. Final simplification12.0

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.43838900620848 \cdot 10^{-27}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq -3.702260431372761 \cdot 10^{-236}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{elif}\;t \leq 5.463357865450902 \cdot 10^{-189}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\ell \cdot \sqrt{\frac{2}{x \cdot x} + \frac{2}{x}}}\\ \mathbf{elif}\;t \leq 1.2822754990186077 \cdot 10^{+34}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{2 \cdot \frac{{\ell}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{x} + 2 \cdot {t}^{2}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 \cdot \frac{x}{x - 1} + 2 \cdot \frac{1}{x - 1}}}\\ \end{array}\]

Reproduce

herbie shell --seed 2021098 
(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)))))