Toniolo and Linder, Equation (7)

Average Error: 42.8 → 9.3

Time: 14.9s

Precision: 64

• Falling back on racket egraph because egg-herbie package not installed

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;t \le -5.6277721314429949 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -9.4314582946777516 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -6.3448789190600404 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 3.34251302474556348 \cdot 10^{119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

C

double f(double x, double l, double t) {
        double r37305 = 2.0;
        double r37306 = sqrt(r37305);
        double r37307 = t;
        double r37308 = r37306 * r37307;
        double r37309 = x;
        double r37310 = 1.0;
        double r37311 = r37309 + r37310;
        double r37312 = r37309 - r37310;
        double r37313 = r37311 / r37312;
        double r37314 = l;
        double r37315 = r37314 * r37314;
        double r37316 = r37307 * r37307;
        double r37317 = r37305 * r37316;
        double r37318 = r37315 + r37317;
        double r37319 = r37313 * r37318;
        double r37320 = r37319 - r37315;
        double r37321 = sqrt(r37320);
        double r37322 = r37308 / r37321;
        return r37322;
}

↓

double f(double x, double l, double t) {
        double r37323 = t;
        double r37324 = -5.627772131442995e+125;
        bool r37325 = r37323 <= r37324;
        double r37326 = 2.0;
        double r37327 = sqrt(r37326);
        double r37328 = r37327 * r37323;
        double r37329 = 3.0;
        double r37330 = pow(r37327, r37329);
        double r37331 = x;
        double r37332 = 2.0;
        double r37333 = pow(r37331, r37332);
        double r37334 = r37330 * r37333;
        double r37335 = r37323 / r37334;
        double r37336 = r37327 * r37333;
        double r37337 = r37323 / r37336;
        double r37338 = r37335 - r37337;
        double r37339 = r37326 * r37338;
        double r37340 = r37327 * r37331;
        double r37341 = r37323 / r37340;
        double r37342 = r37326 * r37341;
        double r37343 = r37323 * r37327;
        double r37344 = r37342 + r37343;
        double r37345 = r37339 - r37344;
        double r37346 = r37328 / r37345;
        double r37347 = -9.431458294677752e-250;
        bool r37348 = r37323 <= r37347;
        double r37349 = sqrt(r37327);
        double r37350 = r37349 * r37323;
        double r37351 = r37349 * r37350;
        double r37352 = pow(r37323, r37332);
        double r37353 = r37326 * r37352;
        double r37354 = l;
        double r37355 = fabs(r37354);
        double r37356 = r37355 / r37331;
        double r37357 = r37355 * r37356;
        double r37358 = r37326 * r37357;
        double r37359 = 4.0;
        double r37360 = r37352 / r37331;
        double r37361 = r37359 * r37360;
        double r37362 = r37358 + r37361;
        double r37363 = r37353 + r37362;
        double r37364 = sqrt(r37363);
        double r37365 = r37351 / r37364;
        double r37366 = -6.34487891906004e-289;
        bool r37367 = r37323 <= r37366;
        double r37368 = r37335 - r37341;
        double r37369 = r37326 * r37368;
        double r37370 = r37369 - r37343;
        double r37371 = r37328 / r37370;
        double r37372 = 3.3425130247455635e+119;
        bool r37373 = r37323 <= r37372;
        double r37374 = r37328 / r37364;
        double r37375 = r37337 + r37341;
        double r37376 = r37326 * r37375;
        double r37377 = r37326 * r37335;
        double r37378 = r37328 - r37377;
        double r37379 = r37376 + r37378;
        double r37380 = r37328 / r37379;
        double r37381 = r37373 ? r37374 : r37380;
        double r37382 = r37367 ? r37371 : r37381;
        double r37383 = r37348 ? r37365 : r37382;
        double r37384 = r37325 ? r37346 : r37383;
        return r37384;
}

Error

Bits error versus x

Bits error versus l

Bits error versus t

Derivation

1. Split input into 5 regimes

2. if t < -5.627772131442995e+125

   1. Initial program 54.5

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

   2. Taylor expanded around -inf 2.1

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

   3. Simplified 2.1

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

   if -5.627772131442995e+125 < t < -9.431458294677752e-250

   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 16.0

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

   4. Applied *-un-lft-identity 16.0

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

   5. Applied add-sqr-sqrt 16.0

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

   6. Applied times-frac 16.0

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

   7. Simplified 16.0

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

   8. Simplified 11.3

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

   10. Applied add-sqr-sqrt 11.3

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

   11. Applied sqrt-prod 11.5

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

   12. Applied associate-*l* 11.4

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

   if -9.431458294677752e-250 < t < -6.34487891906004e-289

   1. Initial program 63.1

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

   2. Taylor expanded around inf 28.6

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

   4. Applied *-un-lft-identity 28.6

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

   5. Applied add-sqr-sqrt 28.6

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

   6. Applied times-frac 28.6

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

   7. Simplified 28.6

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

   8. Simplified 27.6

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

   9. Taylor expanded around -inf 42.7

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

   10. Simplified 42.7

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

   if -6.34487891906004e-289 < t < 3.3425130247455635e+119

   1. Initial program 37.1

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

   2. Taylor expanded around inf 17.3

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

   4. Applied *-un-lft-identity 17.3

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

   5. Applied add-sqr-sqrt 17.3

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

   6. Applied times-frac 17.3

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

   7. Simplified 17.3

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

   8. Simplified 13.3

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

   if 3.3425130247455635e+119 < t

   1. Initial program 54.0

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

   2. Taylor expanded around inf 2.3

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

   3. Simplified 2.3

      \[\leadsto \frac{\sqrt{2} \cdot t}{\color{blue}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}}\]
3. Recombined 5 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le -5.6277721314429949 \cdot 10^{125}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot {x}^{2}}\right) - \left(2 \cdot \frac{t}{\sqrt{2} \cdot x} + t \cdot \sqrt{2}\right)}\\ \mathbf{elif}\;t \le -9.4314582946777516 \cdot 10^{-250}:\\ \;\;\;\;\frac{\sqrt{\sqrt{2}} \cdot \left(\sqrt{\sqrt{2}} \cdot t\right)}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{elif}\;t \le -6.3448789190600404 \cdot 10^{-289}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}} - \frac{t}{\sqrt{2} \cdot x}\right) - t \cdot \sqrt{2}}\\ \mathbf{elif}\;t \le 3.34251302474556348 \cdot 10^{119}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{\sqrt{2 \cdot {t}^{2} + \left(2 \cdot \left(\left|\ell\right| \cdot \frac{\left|\ell\right|}{x}\right) + 4 \cdot \frac{{t}^{2}}{x}\right)}}\\ \mathbf{else}:\\ \;\;\;\;\frac{\sqrt{2} \cdot t}{2 \cdot \left(\frac{t}{\sqrt{2} \cdot {x}^{2}} + \frac{t}{\sqrt{2} \cdot x}\right) + \left(\sqrt{2} \cdot t - 2 \cdot \frac{t}{{\left(\sqrt{2}\right)}^{3} \cdot {x}^{2}}\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2020042 
(FPCore (x l t)
  :name "Toniolo and Linder, Equation (7)"
  :precision binary64
  (/ (* (sqrt 2) t) (sqrt (- (* (/ (+ x 1) (- x 1)) (+ (* l l) (* 2 (* t t)))) (* l l)))))