Average Error: 42.9 → 9.3
Time: 9.4s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -4.775489866798947 \cdot 10^{+107}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 8.323497236838193 \cdot 10^{-203}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}} \cdot \left(\sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}}\right)\right)}}\\ \mathbf{elif}\;t \leq 7.461103816305763 \cdot 10^{-157} \lor \neg \left(t \leq 9.712703512221514 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \end{array}\]
\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 \leq -4.775489866798947 \cdot 10^{+107}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\

\mathbf{elif}\;t \leq 8.323497236838193 \cdot 10^{-203}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}} \cdot \left(\sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}}\right)\right)}}\\

\mathbf{elif}\;t \leq 7.461103816305763 \cdot 10^{-157} \lor \neg \left(t \leq 9.712703512221514 \cdot 10^{+113}\right):\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\

\mathbf{else}:\\
\;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\

\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 -4.775489866798947e+107)
   (/
    (* t (sqrt 2.0))
    (-
     (/ t (* (sqrt 2.0) (* x x)))
     (+ (* t (sqrt 2.0)) (* (/ 2.0 (sqrt 2.0)) (+ (/ t (* x x)) (/ t x))))))
   (if (<= t 8.323497236838193e-203)
     (/
      (* t (sqrt 2.0))
      (sqrt
       (+
        (* 4.0 (/ (* t t) x))
        (*
         2.0
         (*
          (sqrt (+ (* t t) (/ l (/ x l))))
          (*
           (sqrt (sqrt (+ (* t t) (/ l (/ x l)))))
           (sqrt (sqrt (+ (* t t) (/ l (/ x l)))))))))))
     (if (or (<= t 7.461103816305763e-157) (not (<= t 9.712703512221514e+113)))
       (/
        (* t (sqrt 2.0))
        (+
         (* (/ 2.0 (sqrt 2.0)) (+ (/ t (* x x)) (/ t x)))
         (- (* t (sqrt 2.0)) (/ t (* (sqrt 2.0) (* x x))))))
       (/
        (* t (sqrt 2.0))
        (sqrt (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (/ l (/ x l)))))))))))
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 <= -4.775489866798947e+107) {
		tmp = (t * sqrt(2.0)) / ((t / (sqrt(2.0) * (x * x))) - ((t * sqrt(2.0)) + ((2.0 / sqrt(2.0)) * ((t / (x * x)) + (t / x)))));
	} else if (t <= 8.323497236838193e-203) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * (sqrt((t * t) + (l / (x / l))) * (sqrt(sqrt((t * t) + (l / (x / l)))) * sqrt(sqrt((t * t) + (l / (x / l))))))));
	} else if ((t <= 7.461103816305763e-157) || !(t <= 9.712703512221514e+113)) {
		tmp = (t * sqrt(2.0)) / (((2.0 / sqrt(2.0)) * ((t / (x * x)) + (t / x))) + ((t * sqrt(2.0)) - (t / (sqrt(2.0) * (x * x)))));
	} else {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l / (x / l)))));
	}
	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 < -4.7754898667989469e107

    1. Initial program 52.3

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

    2. Taylor expanded around -inf 2.2

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

    3. Simplified2.2

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

    if -4.7754898667989469e107 < t < 8.32349723683819328e-203

    1. Initial program 41.8

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

    2. Taylor expanded around inf 20.6

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

    3. Simplified20.6

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

    5. Applied associate-/l*_binary64_2416.8

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

    7. Applied add-sqr-sqrt_binary64_9816.8

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

    9. Applied add-sqr-sqrt_binary64_9816.8

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

    if 8.32349723683819328e-203 < t < 7.46110381630576257e-157 or 9.71270351222151374e113 < t

    1. Initial program 53.5

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

    2. Taylor expanded around inf 6.5

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

    3. Simplified6.5

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

    if 7.46110381630576257e-157 < t < 9.71270351222151374e113

    1. Initial program 26.1

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

    2. Taylor expanded around inf 10.4

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

    3. Simplified10.4

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

    5. Applied associate-/l*_binary64_245.6

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

  4. Final simplification9.3

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.775489866798947 \cdot 10^{+107}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 8.323497236838193 \cdot 10^{-203}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}} \cdot \left(\sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}} \cdot \sqrt{\sqrt{t \cdot t + \frac{\ell}{\frac{x}{\ell}}}}\right)\right)}}\\ \mathbf{elif}\;t \leq 7.461103816305763 \cdot 10^{-157} \lor \neg \left(t \leq 9.712703512221514 \cdot 10^{+113}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020288 
(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)))))