Toniolo and Linder, Equation (7)

Average Error: 43.3 → 11.1

Time: 13.6s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}\\ t_2 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -1.7789831318513475 \cdot 10^{-12}:\\ \;\;\;\;\frac{t_2}{-t_1}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_3 := \frac{t_2}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{if}\;t \leq 5.746839589054363 \cdot 10^{-281}:\\ \;\;\;\;t_3\\ \mathbf{elif}\;t \leq 4.394123494096578 \cdot 10^{-200}:\\ \;\;\;\;\begin{array}{l} t_4 := \sqrt{2} \cdot x\\ \frac{t_2}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_4}, \frac{\ell \cdot \ell}{t \cdot t_4}\right)\right)} \end{array}\\ \mathbf{elif}\;t \leq 3.4016621676448906 \cdot 10^{+73}:\\ \;\;\;\;t_3\\ \mathbf{else}:\\ \;\;\;\;\frac{t_2}{t_1}\\ \end{array}\\ \end{array} \]

FPCore

(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 (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0)))))))
        (t_2 (* t (sqrt 2.0))))
   (if (<= t -1.7789831318513475e-12)
     (/ t_2 (- t_1))
     (let* ((t_3
             (/
              t_2
              (sqrt
               (fma 4.0 (/ (* t t) x) (* 2.0 (+ (* t t) (/ (* l l) x))))))))
       (if (<= t 5.746839589054363e-281)
         t_3
         (if (<= t 4.394123494096578e-200)
           (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 3.4016621676448906e+73) t_3 (/ t_2 t_1))))))))

C

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 = t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0))));
	double t_2 = t * sqrt(2.0);
	double tmp;
	if (t <= -1.7789831318513475e-12) {
		tmp = t_2 / -t_1;
	} else {
		double t_3 = t_2 / sqrt(fma(4.0, ((t * t) / x), (2.0 * ((t * t) + ((l * l) / x)))));
		double tmp_1;
		if (t <= 5.746839589054363e-281) {
			tmp_1 = t_3;
		} else if (t <= 4.394123494096578e-200) {
			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 <= 3.4016621676448906e+73) {
			tmp_1 = t_3;
		} else {
			tmp_1 = t_2 / t_1;
		}
		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 < -1.7789831318513475e-12

   1. Initial program 41.5

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

   2. Simplified 41.5

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

   3. Taylor expanded in t around -inf 5.2

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

   4. Simplified 5.2

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

   if -1.7789831318513475e-12 < t < 5.74683958905436263e-281 or 4.39412349409657786e-200 < t < 3.40166216764489058e73

   1. Initial program 40.3

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

   2. Simplified 40.3

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

   3. Taylor expanded in x around inf 17.9

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

   4. Simplified 17.9

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

   if 5.74683958905436263e-281 < t < 4.39412349409657786e-200

   1. Initial program 62.5

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

   2. Simplified 62.5

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

   3. Taylor expanded in x around inf 38.7

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

   4. Simplified 38.7

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

   5. Taylor expanded in x around inf 23.6

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

   6. Simplified 23.6

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

   if 3.40166216764489058e73 < t

   1. Initial program 46.6

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

   2. Simplified 46.6

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

   3. Taylor expanded in t around inf 2.9

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

   4. Simplified 2.9

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

4. Final simplification 11.1

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -1.7789831318513475 \cdot 10^{-12}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 5.746839589054363 \cdot 10^{-281}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 4.394123494096578 \cdot 10^{-200}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 3.4016621676448906 \cdot 10^{+73}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022097 
(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)))))