Toniolo and Linder, Equation (7)

Average Error: 43.5 → 10.5

Time: 9.5s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \leq -2.650489049077127 \cdot 10^{+17}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq -7.168543711359687 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -3.8356938576006817 \cdot 10^{-262}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{\sqrt{2} \cdot x} - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.2532264623485196 \cdot 10^{-241}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{1}{\frac{x}{\ell \cdot \ell}}\right)}}\\ \mathbf{elif}\;t \leq 1.7732721537111569 \cdot 10^{-180} \lor \neg \left(t \leq 1.0360494278012444 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - \frac{\frac{t}{x}}{\sqrt{2} \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \end{array}\]

FPCore

(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 -2.650489049077127e+17)
   (/
    (* 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 -7.168543711359687e-209)
     (/
      (* t (sqrt 2.0))
      (sqrt (+ (* 4.0 (/ (* t t) x)) (* 2.0 (+ (* t t) (* l (/ l x)))))))
     (if (<= t -3.8356938576006817e-262)
       (/
        (* t (sqrt 2.0))
        (-
         (/ (/ t x) (* (sqrt 2.0) x))
         (+ (* t (sqrt 2.0)) (* 2.0 (/ t (* (sqrt 2.0) x))))))
       (if (<= t 1.2532264623485196e-241)
         (/
          (* t (sqrt 2.0))
          (sqrt
           (+
            (* 4.0 (/ (* t t) x))
            (* 2.0 (+ (* t t) (/ 1.0 (/ x (* l l))))))))
         (if (or (<= t 1.7732721537111569e-180)
                 (not (<= t 1.0360494278012444e+30)))
           (/
            (* t (sqrt 2.0))
            (-
             (+ (* t (sqrt 2.0)) (* 2.0 (/ t (* (sqrt 2.0) x))))
             (/ (/ t x) (* (sqrt 2.0) x))))
           (/
            (* t (sqrt 2.0))
            (sqrt
             (+
              (* 4.0 (/ (* t t) x))
              (* 2.0 (+ (* t t) (/ l (/ x l)))))))))))))

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 tmp;
	if (t <= -2.650489049077127e+17) {
		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 <= -7.168543711359687e-209) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (l * (l / x)))));
	} else if (t <= -3.8356938576006817e-262) {
		tmp = (t * sqrt(2.0)) / (((t / x) / (sqrt(2.0) * x)) - ((t * sqrt(2.0)) + (2.0 * (t / (sqrt(2.0) * x)))));
	} else if (t <= 1.2532264623485196e-241) {
		tmp = (t * sqrt(2.0)) / sqrt((4.0 * ((t * t) / x)) + (2.0 * ((t * t) + (1.0 / (x / (l * l))))));
	} else if ((t <= 1.7732721537111569e-180) || !(t <= 1.0360494278012444e+30)) {
		tmp = (t * sqrt(2.0)) / (((t * sqrt(2.0)) + (2.0 * (t / (sqrt(2.0) * x)))) - ((t / x) / (sqrt(2.0) * 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

Derivation

1. Split input into 6 regimes

2. if t < -265048904907712700

   1. Initial program 42.9

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

   2. Taylor expanded around -inf 4.7

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

   3. Simplified 4.7

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

   if -265048904907712700 < t < -7.16854371135968704e-209

   1. Initial program 36.8

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

   2. Taylor expanded around inf 14.6

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

   3. Simplified 14.6

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

   5. Applied *-un-lft-identity_binary64_78 14.6

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

   6. Applied times-frac_binary64_84 10.3

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

   7. Simplified 10.3

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

   if -7.16854371135968704e-209 < t < -3.83569385760068168e-262

   1. Initial program 62.7

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

   2. Taylor expanded around inf 31.4

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

   3. Simplified 31.4

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

   4. Taylor expanded around -inf 38.7

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

   5. Simplified 38.7

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

   if -3.83569385760068168e-262 < t < 1.25322646234851964e-241

   1. Initial program 63.2

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

   2. Taylor expanded around inf 31.1

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

   3. Simplified 31.1

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

   5. Applied clear-num_binary64_77 31.5

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

   if 1.25322646234851964e-241 < t < 1.77327215371115688e-180 or 1.0360494278012444e30 < t

   1. Initial program 46.1

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

   2. Taylor expanded around inf 40.4

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

   3. Simplified 40.4

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

   4. Taylor expanded around inf 9.2

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

   5. Simplified 9.2

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

   if 1.77327215371115688e-180 < t < 1.0360494278012444e30

   1. Initial program 33.4

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

   2. Taylor expanded around inf 13.3

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

   3. Simplified 13.3

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

   5. Applied associate-/l*_binary64_23 7.6

      \[\leadsto \frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \color{blue}{\frac{\ell}{\frac{x}{\ell}}}\right)}}\]
3. Recombined 6 regimes into one program.

4. Final simplification 10.5

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \leq -2.650489049077127 \cdot 10^{+17}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{t}{\sqrt{2} \cdot \left(x \cdot x\right)} - \left(t \cdot \sqrt{2} + \frac{2}{\sqrt{2}} \cdot \left(\frac{t}{x \cdot x} + \frac{t}{x}\right)\right)}\\ \mathbf{elif}\;t \leq -7.168543711359687 \cdot 10^{-209}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \leq -3.8356938576006817 \cdot 10^{-262}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\frac{\frac{t}{x}}{\sqrt{2} \cdot x} - \left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right)}\\ \mathbf{elif}\;t \leq 1.2532264623485196 \cdot 10^{-241}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{1}{\frac{x}{\ell \cdot \ell}}\right)}}\\ \mathbf{elif}\;t \leq 1.7732721537111569 \cdot 10^{-180} \lor \neg \left(t \leq 1.0360494278012444 \cdot 10^{+30}\right):\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\left(t \cdot \sqrt{2} + 2 \cdot \frac{t}{\sqrt{2} \cdot x}\right) - \frac{\frac{t}{x}}{\sqrt{2} \cdot x}}\\ \mathbf{else}:\\ \;\;\;\;\frac{t \cdot \sqrt{2}}{\sqrt{4 \cdot \frac{t \cdot t}{x} + 2 \cdot \left(t \cdot t + \frac{\ell}{\frac{x}{\ell}}\right)}}\\ \end{array}\]

Reproduce

herbie shell --seed 2020356 
(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)))))