Average Error: 43.0 → 9.7
Time: 9.5s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}}\]
↓
\[\begin{array}{l} \mathbf{if}\;t \leq -6.640054786982412 \cdot 10^{+146}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq -1.0408412389809283 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -1.1072730107488086 \cdot 10^{-286}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2639360950810393 \cdot 10^{-296}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq 6.794026911200415 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 4.430742223517365 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \end{array}\]
\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 -6.640054786982412 \cdot 10^{+146}:\\
\;\;\;\;\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 -1.0408412389809283 \cdot 10^{-243}:\\
\;\;\;\;\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)}}\\

\mathbf{elif}\;t \leq -1.1072730107488086 \cdot 10^{-286}:\\
\;\;\;\;\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 2.2639360950810393 \cdot 10^{-296}:\\
\;\;\;\;\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)}}\\

\mathbf{elif}\;t \leq 6.794026911200415 \cdot 10^{-186}:\\
\;\;\;\;\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{elif}\;t \leq 4.430742223517365 \cdot 10^{+61}:\\
\;\;\;\;\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)}}\\

\mathbf{else}:\\
\;\;\;\;\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)}\\

\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 -6.640054786982412e+146)
   (/
    (* 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 -1.0408412389809283e-243)
     (/
      (* t (sqrt 2.0))
      (sqrt (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (/ l (/ x l)))))))
     (if (<= t -1.1072730107488086e-286)
       (/
        (* 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 2.2639360950810393e-296)
         (/
          (* t (sqrt 2.0))
          (sqrt (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (/ l (/ x l)))))))
         (if (<= t 6.794026911200415e-186)
           (/
            (* t (sqrt 2.0))
            (+
             (* (/ 2.0 (sqrt 2.0)) (+ (/ t (* x x)) (/ t x)))
             (- (* t (sqrt 2.0)) (/ t (* (sqrt 2.0) (* x x))))))
           (if (<= t 4.430742223517365e+61)
             (/
              (* t (sqrt 2.0))
              (sqrt
               (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (/ l (/ x l)))))))
             (/
              (* t (sqrt 2.0))
              (+
               (* (/ 2.0 (sqrt 2.0)) (+ (/ t (* x x)) (/ t x)))
               (- (* t (sqrt 2.0)) (/ t (* (sqrt 2.0) (* x 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 <= -6.640054786982412e+146) {
		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 <= -1.0408412389809283e-243) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l / (x / l)))));
	} else if (t <= -1.1072730107488086e-286) {
		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 <= 2.2639360950810393e-296) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l / (x / l)))));
	} else if (t <= 6.794026911200415e-186) {
		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 if (t <= 4.430742223517365e+61) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l / (x / l)))));
	} else {
		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)))));
	}
	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 3 regimes

  2. if t < -6.64005478698241218e146 or -1.0408412389809283e-243 < t < -1.10727301074880864e-286

    1. Initial program 61.1

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

    2. Taylor expanded around -inf 7.9

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

    3. Simplified7.9

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

    if -6.64005478698241218e146 < t < -1.0408412389809283e-243 or -1.10727301074880864e-286 < t < 2.2639360950810393e-296 or 6.7940269112004151e-186 < t < 4.43074222351736488e61

    1. Initial program 33.0

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

    2. Taylor expanded around inf 14.7

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

    3. Simplified14.7

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

    5. Applied associate-/l*_binary649.9

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

    if 2.2639360950810393e-296 < t < 6.7940269112004151e-186 or 4.43074222351736488e61 < t

    1. Initial program 49.0

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

    2. Taylor expanded around inf 10.6

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

    3. Simplified10.6

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

  4. Final simplification9.7

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.640054786982412 \cdot 10^{+146}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq -1.0408412389809283 \cdot 10^{-243}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq -1.1072730107488086 \cdot 10^{-286}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq 2.2639360950810393 \cdot 10^{-296}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{elif}\;t \leq 6.794026911200415 \cdot 10^{-186}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \mathbf{elif}\;t \leq 4.430742223517365 \cdot 10^{+61}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right) + \left(t \cdot \sqrt{2} - \frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020273 
(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)))))