Toniolo and Linder, Equation (7)

Average Error: 43.8 → 10.6

Time: 28.7s

Precision: binary64

Math

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

↓

\[\begin{array}{l} t_1 := t \cdot \sqrt{2}\\ \mathbf{if}\;t \leq -4.307136892331502 \cdot 10^{+72}:\\ \;\;\;\;\frac{t_1}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{else}:\\ \;\;\;\;\begin{array}{l} t_2 := \frac{t_1}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{if}\;t \leq 8.04365893904582 \cdot 10^{-301}:\\ \;\;\;\;t_2\\ \mathbf{elif}\;t \leq 1.5164001531321383 \cdot 10^{-162}:\\ \;\;\;\;\begin{array}{l} t_3 := \sqrt{2} \cdot x\\ \frac{t_1}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{t_3}, \frac{\ell \cdot \ell}{t \cdot t_3}\right)\right)} \end{array}\\ \mathbf{elif}\;t \leq 1.659580308807747 \cdot 10^{+47}:\\ \;\;\;\;t_2\\ \mathbf{else}:\\ \;\;\;\;\frac{t_1}{t \cdot \sqrt{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}}\\ \end{array}\\ \end{array} \]

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))))
   (if (<= t -4.307136892331502e+72)
     (/ t_1 (- (* t (sqrt (+ (/ 2.0 (- x 1.0)) (* 2.0 (/ x (- x 1.0))))))))
     (let* ((t_2
             (/
              t_1
              (sqrt
               (fma 4.0 (/ (* t t) x) (* 2.0 (+ (* t t) (/ (* l l) x))))))))
       (if (<= t 8.04365893904582e-301)
         t_2
         (if (<= t 1.5164001531321383e-162)
           (let* ((t_3 (* (sqrt 2.0) x)))
             (/
              t_1
              (fma t (sqrt 2.0) (fma 2.0 (/ t t_3) (/ (* l l) (* t t_3))))))
           (if (<= t 1.659580308807747e+47)
             t_2
             (/
              t_1
              (*
               t
               (sqrt
                (+
                 2.0
                 (+
                  (/ 4.0 x)
                  (+ (/ 4.0 (pow x 3.0)) (/ 4.0 (* x x)))))))))))))))

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);
	double tmp;
	if (t <= -4.307136892331502e+72) {
		tmp = t_1 / -(t * sqrt((2.0 / (x - 1.0)) + (2.0 * (x / (x - 1.0)))));
	} else {
		double t_2 = t_1 / sqrt(fma(4.0, ((t * t) / x), (2.0 * ((t * t) + ((l * l) / x)))));
		double tmp_1;
		if (t <= 8.04365893904582e-301) {
			tmp_1 = t_2;
		} else if (t <= 1.5164001531321383e-162) {
			double t_3 = sqrt(2.0) * x;
			tmp_1 = t_1 / fma(t, sqrt(2.0), fma(2.0, (t / t_3), ((l * l) / (t * t_3))));
		} else if (t <= 1.659580308807747e+47) {
			tmp_1 = t_2;
		} else {
			tmp_1 = t_1 / (t * sqrt(2.0 + ((4.0 / x) + ((4.0 / pow(x, 3.0)) + (4.0 / (x * 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 < -4.30713689233150193e72

   1. Initial program 48.0

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

   2. Simplified 48.0

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

   3. Taylor expanded in t around -inf 3.1

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

   4. Simplified 3.1

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

   if -4.30713689233150193e72 < t < 8.04365893904582e-301 or 1.5164001531321383e-162 < t < 1.65958030880774699e47

   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. Simplified 37.2

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

   3. Taylor expanded in x around inf 15.3

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

   4. Simplified 15.3

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

   if 8.04365893904582e-301 < t < 1.5164001531321383e-162

   1. Initial program 62.8

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

   2. Simplified 62.8

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

   3. Taylor expanded in x around inf 24.9

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

   4. Simplified 24.9

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

   if 1.65958030880774699e47 < t

   1. Initial program 44.5

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

   2. Simplified 44.5

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

   3. Taylor expanded in x around inf 43.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{4 \cdot \frac{{t}^{2}}{{x}^{2}} + \left(4 \cdot \frac{{t}^{2}}{x} + \left(4 \cdot \frac{{t}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{2}} + \left(2 \cdot \frac{{\ell}^{2}}{{x}^{3}} + \left(2 \cdot \frac{{\ell}^{2}}{x} + 2 \cdot {t}^{2}\right)\right)\right)\right)\right)}}} \]

   4. Simplified 43.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{\color{blue}{\mathsf{fma}\left(4, \frac{t \cdot t}{x \cdot x}, \mathsf{fma}\left(4, \frac{t \cdot t}{x}, \mathsf{fma}\left(4, \frac{t \cdot t}{{x}^{3}}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{x \cdot x}, \mathsf{fma}\left(2, \frac{\ell \cdot \ell}{{x}^{3}}, 2 \cdot \left(\frac{\ell \cdot \ell}{x} + t \cdot t\right)\right)\right)\right)\right)\right)}}} \]

   5. Taylor expanded in t around inf 4.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{t \cdot \sqrt{2 + \left(4 \cdot \frac{1}{x} + \left(4 \cdot \frac{1}{{x}^{3}} + 4 \cdot \frac{1}{{x}^{2}}\right)\right)}}} \]

   6. Simplified 4.3

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

4. Final simplification 10.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -4.307136892331502 \cdot 10^{+72}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{-t \cdot \sqrt{\frac{2}{x - 1} + 2 \cdot \frac{x}{x - 1}}}\\ \mathbf{elif}\;t \leq 8.04365893904582 \cdot 10^{-301}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{elif}\;t \leq 1.5164001531321383 \cdot 10^{-162}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\mathsf{fma}\left(t, \sqrt{2}, \mathsf{fma}\left(2, \frac{t}{\sqrt{2} \cdot x}, \frac{\ell \cdot \ell}{t \cdot \left(\sqrt{2} \cdot x\right)}\right)\right)}\\ \mathbf{elif}\;t \leq 1.659580308807747 \cdot 10^{+47}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{\mathsf{fma}\left(4, \frac{t \cdot t}{x}, 2 \cdot \left(t \cdot t + \frac{\ell \cdot \ell}{x}\right)\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{t \cdot \sqrt{2 + \left(\frac{4}{x} + \left(\frac{4}{{x}^{3}} + \frac{4}{x \cdot x}\right)\right)}}\\ \end{array} \]

Reproduce

herbie shell --seed 2022067 
(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)))))