Toniolo and Linder, Equation (7)

Average Error: 43.1 → 9.6

Time: 13.1s

Precision: binary64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -6.68023045957239488 \cdot 10^{30}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -1.7722596374613524 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le -1.3583216648888047 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.3270366640675283 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 6.0473137750788973 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 2.9843805005488113 \cdot 10^{81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

C

double code(double x, double l, double t) {
	return ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (((double) (((double) (x + 1.0)) / ((double) (x - 1.0)))) * ((double) (((double) (l * l)) + ((double) (2.0 * ((double) (t * t)))))))) - ((double) (l * l))))))));
}

↓

double code(double x, double l, double t) {
	double VAR;
	if ((t <= -6.680230459572395e+30)) {
		VAR = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0)))))) - ((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))))))) - ((double) (((double) sqrt(2.0)) * t)))) - ((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x))))))))));
	} else {
		double VAR_1;
		if ((t <= -1.7722596374613524e-241)) {
			VAR_1 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x))))))))))))));
		} else {
			double VAR_2;
			if ((t <= -1.3583216648888047e-298)) {
				VAR_2 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0)))))) - ((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))))))) - ((double) (((double) sqrt(2.0)) * t)))) - ((double) (2.0 * ((double) (t / ((double) (((double) sqrt(2.0)) * x))))))))));
			} else {
				double VAR_3;
				if ((t <= 3.3270366640675283e-243)) {
					VAR_3 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x))))))))))))));
				} else {
					double VAR_4;
					if ((t <= 6.047313775078897e-204)) {
						VAR_4 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))) + ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))))) + ((double) (((double) (((double) sqrt(2.0)) * t)) - ((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0))))))))))))));
					} else {
						double VAR_5;
						if ((t <= 2.9843805005488113e+81)) {
							VAR_5 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) sqrt(((double) (((double) (4.0 * ((double) (((double) pow(t, 2.0)) / x)))) + ((double) (2.0 * ((double) (((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x)))))))) * ((double) sqrt(((double) (((double) pow(t, 2.0)) + ((double) (l * ((double) (l / x))))))))))))))))));
						} else {
							VAR_5 = ((double) (((double) (((double) sqrt(2.0)) * t)) / ((double) (((double) (2.0 * ((double) (((double) (t / ((double) (((double) sqrt(2.0)) * ((double) pow(x, 2.0)))))) + ((double) (t / ((double) (((double) sqrt(2.0)) * x)))))))) + ((double) (((double) (((double) sqrt(2.0)) * t)) - ((double) (2.0 * ((double) (t / ((double) (((double) pow(((double) sqrt(2.0)), 3.0)) * ((double) pow(x, 2.0))))))))))))));
						}
						VAR_4 = VAR_5;
					}
					VAR_3 = VAR_4;
				}
				VAR_2 = VAR_3;
			}
			VAR_1 = VAR_2;
		}
		VAR = VAR_1;
	}
	return VAR;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 4 regimes

2. if t < -6.68023045957239488e30 or -1.7722596374613524e-241 < t < -1.3583216648888047e-298

   1. Initial program 46.3

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

   2. Taylor expanded around -inf 8.3

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

   3. Simplified 8.3

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

   if -6.68023045957239488e30 < t < -1.7722596374613524e-241 or -1.3583216648888047e-298 < t < 3.3270366640675283e-243

   1. Initial program 43.4

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

   2. Taylor expanded around inf 19.3

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

   3. Simplified 19.3

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

   5. Applied *-un-lft-identity 19.3

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

   6. Applied add-sqr-sqrt 42.3

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

   7. Applied unpow-prod-down 42.3

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

   8. Applied times-frac 40.7

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

   9. Simplified 40.6

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

   10. Simplified 15.6

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

   if 3.3270366640675283e-243 < t < 6.0473137750788973e-204 or 2.9843805005488113e81 < t

   1. Initial program 49.1

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

   2. Taylor expanded around inf 6.1

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

   3. Simplified 6.1

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

   if 6.0473137750788973e-204 < t < 2.9843805005488113e81

   1. Initial program 31.6

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

   2. Taylor expanded around inf 13.9

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

   3. Simplified 13.9

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

   5. Applied *-un-lft-identity 13.9

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

   6. Applied add-sqr-sqrt 38.7

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

   7. Applied unpow-prod-down 38.7

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

   8. Applied times-frac 36.4

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

   9. Simplified 36.4

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

   10. Simplified 8.9

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

   12. Applied add-sqr-sqrt 8.9

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

4. Final simplification 9.6

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -6.68023045957239488 \cdot 10^{30}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le -1.7722596374613524 \cdot 10^{-241}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le -1.3583216648888047 \cdot 10^{-298}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\left(2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \sqrt{2} \cdot t\right) - 2 \cdot \frac{t}{\sqrt{2} \cdot x}}\\ \mathbf{elif}\;t \le 3.3270366640675283 \cdot 10^{-243}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left({t}^{2} + \ell \cdot \frac{\ell}{x}\right)}}\\ \mathbf{elif}\;t \le 6.0473137750788973 \cdot 10^{-204}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \mathbf{elif}\;t \le 2.9843805005488113 \cdot 10^{81}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{4 \cdot \frac{{t}^{2}}{x} + 2 \cdot \left(\sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}} \cdot \sqrt{{t}^{2} + \ell \cdot \frac{\ell}{x}}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020162 
(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)))))