Average Error: 42.7 → 11.1
Time: 15.6s
Precision: binary64
\[\frac{\sqrt{2} \cdot t}{\sqrt{\frac{x + 1}{x - 1} \cdot \left(\ell \cdot \ell + 2 \cdot \left(t \cdot t\right)\right) - \ell \cdot \ell}} \]
↓
\[\begin{array}{l} t_1 := \frac{x}{-1 + x}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -6.790049380025751 \cdot 10^{-65}:\\ \;\;\;\;\frac{t_2}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot t_1}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{t_2}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{if}\;t \leq 1.6105793316811791 \cdot 10^{-264}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 3.3813144302125017 \cdot 10^{-122}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{2} \cdot x\\ \frac{t_2}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_4}, \frac{\ell \cdot \ell}{t \cdot t_4}\right)\right)} \end{array}\\ \mathbf{elif}\;t \leq 4.810710790881342 \cdot 10^{-47}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t \cdot \sqrt{\mathsf{fma}\left(2, t_1, \frac{2}{-1 + x}\right)}}\\ \end{array}\\ \end{array} \]
\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}
t_1 := \frac{x}{-1 + x}\\
t_2 := t \cdot \sqrt{2}\\
\mathbf{if}\;t \leq -6.790049380025751 \cdot 10^{-65}:\\
\;\;\;\;\frac{t_2}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot t_1}}\\

\mathbf{else}:\\
\;\;\;\;\begin{array}{l}
t_3 := \frac{t_2}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\
\mathbf{if}\;t \leq 1.6105793316811791 \cdot 10^{-264}:\\
\;\;\;\;t_3\\

\mathbf{elif}\;t \leq 3.3813144302125017 \cdot 10^{-122}:\\
\;\;\;\;\begin{array}{l}
t_4 := \sqrt{2} \cdot x\\
\frac{t_2}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_4}, \frac{\ell \cdot \ell}{t \cdot t_4}\right)\right)}
\end{array}\\

\mathbf{elif}\;t \leq 4.810710790881342 \cdot 10^{-47}:\\
\;\;\;\;t_3\\

\mathbf{else}:\\
\;\;\;\;\frac{t_2}{t \cdot \sqrt{\mathsf{fma}\left(2, t_1, \frac{2}{-1 + x}\right)}}\\


\end{array}\\


\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
 (let* ((t_1 (/ x (+ -1.0 x))) (t_2 (* t (sqrt 2.0))))
   (if (<= t -6.790049380025751e-65)
     (/ t_2 (- (* t (sqrt (+ (* 2.0 (/ 1.0 (+ -1.0 x))) (* 2.0 t_1))))))
     (let* ((t_3
             (/
              t_2
              (sqrt
               (fma 2.0 (+ (/ (* l l) x) (* t t)) (* 4.0 (/ (* t t) x)))))))
       (if (<= t 1.6105793316811791e-264)
         t_3
         (if (<= t 3.3813144302125017e-122)
           (let* ((t_4 (* (sqrt 2.0) x)))
             (/
              t_2
              (fma t (sqrt 2.0) (fma 2.0 (/ t t_4) (/ (* l l) (* t t_4))))))
           (if (<= t 4.810710790881342e-47)
             t_3
             (/ t_2 (* t (sqrt (fma 2.0 t_1 (/ 2.0 (+ -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 t_1 = x / (-1.0 + x);
	double t_2 = t * sqrt(2.0);
	double tmp;
	if (t <= -6.790049380025751e-65) {
		tmp = t_2 / -(t * sqrt((2.0 * (1.0 / (-1.0 + x))) + (2.0 * t_1)));
	} else {
		double t_3 = t_2 / sqrt(fma(2.0, (((l * l) / x) + (t * t)), (4.0 * ((t * t) / x))));
		double tmp_1;
		if (t <= 1.6105793316811791e-264) {
			tmp_1 = t_3;
		} else if (t <= 3.3813144302125017e-122) {
			double t_4 = sqrt(2.0) * x;
			tmp_1 = t_2 / fma(t, sqrt(2.0), fma(2.0, (t / t_4), ((l * l) / (t * t_4))));
		} else if (t <= 4.810710790881342e-47) {
			tmp_1 = t_3;
		} else {
			tmp_1 = t_2 / (t * sqrt(fma(2.0, t_1, (2.0 / (-1.0 + x)))));
		}
		tmp = tmp_1;
	}
	return tmp;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

  1. Split input into 4 regimes

  2. if t < -6.79004938002575122e-65

    1. Initial program 38.5

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

    2. Simplified38.5

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

    3. Taylor expanded in t around -inf 5.9

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

    if -6.79004938002575122e-65 < t < 1.6105793316811791e-264 or 3.38131443021250172e-122 < t < 4.8107107908813422e-47

    1. Initial program 47.5

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

    2. Simplified47.5

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

    3. Taylor expanded in x around inf 20.2

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

    4. Simplified20.2

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

    if 1.6105793316811791e-264 < t < 3.38131443021250172e-122

    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. Simplified55.1

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

    3. Taylor expanded in x around inf 22.6

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

    4. Simplified22.6

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

    if 4.8107107908813422e-47 < t

    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. Simplified39.8

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

    3. Taylor expanded in t around inf 6.2

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

    4. Simplified6.2

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{x - 1}, \frac{2}{x - 1}\right)}}} \]
  3. Recombined 4 regimes into one program.

  4. Final simplification11.1

    \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -6.790049380025751 \cdot 10^{-65}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{2 \cdot \frac{1}{-1 + x} + 2 \cdot \frac{x}{-1 + x}}}\\ \mathbf{elif}\;t \leq 1.6105793316811791 \cdot 10^{-264}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{elif}\;t \leq 3.3813144302125017 \cdot 10^{-122}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 4.810710790881342 \cdot 10^{-47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x} + t \cdot t, 4 \cdot \frac{t \cdot t}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\mathsf{fma}\left(2, \frac{x}{-1 + x}, \frac{2}{-1 + x}\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2021352 
(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)))))