Toniolo and Linder, Equation (13)

Average Error: 35.1 → 30.9

Time: 44.7s

Precision: 64

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

Math

\[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]

↓

\[\begin{array}{l} \mathbf{if}\;t \le 3.130632929319652416005471644268637375687 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} \cdot \left(U \cdot \left(\left(2 \cdot n\right) \cdot \sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;t \le 5.953905810989119972468590963716396861846 \cdot 10^{-86} \lor \neg \left(t \le 2.817007442329380152502379998300985537163 \cdot 10^{107}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r217391 = 2.0;
        double r217392 = n;
        double r217393 = r217391 * r217392;
        double r217394 = U;
        double r217395 = r217393 * r217394;
        double r217396 = t;
        double r217397 = l;
        double r217398 = r217397 * r217397;
        double r217399 = Om;
        double r217400 = r217398 / r217399;
        double r217401 = r217391 * r217400;
        double r217402 = r217396 - r217401;
        double r217403 = r217397 / r217399;
        double r217404 = pow(r217403, r217391);
        double r217405 = r217392 * r217404;
        double r217406 = U_;
        double r217407 = r217394 - r217406;
        double r217408 = r217405 * r217407;
        double r217409 = r217402 - r217408;
        double r217410 = r217395 * r217409;
        double r217411 = sqrt(r217410);
        return r217411;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r217412 = t;
        double r217413 = 3.1306329293196524e-271;
        bool r217414 = r217412 <= r217413;
        double r217415 = U;
        double r217416 = U_;
        double r217417 = r217415 - r217416;
        double r217418 = l;
        double r217419 = Om;
        double r217420 = r217418 / r217419;
        double r217421 = 2.0;
        double r217422 = 2.0;
        double r217423 = r217422 / r217421;
        double r217424 = r217421 * r217423;
        double r217425 = pow(r217420, r217424);
        double r217426 = r217417 * r217425;
        double r217427 = n;
        double r217428 = r217426 * r217427;
        double r217429 = r217419 / r217418;
        double r217430 = r217418 / r217429;
        double r217431 = r217422 * r217430;
        double r217432 = r217428 + r217431;
        double r217433 = r217412 - r217432;
        double r217434 = cbrt(r217433);
        double r217435 = r217422 * r217427;
        double r217436 = r217435 * r217434;
        double r217437 = r217415 * r217436;
        double r217438 = r217434 * r217437;
        double r217439 = r217412 - r217431;
        double r217440 = pow(r217420, r217423);
        double r217441 = r217427 * r217440;
        double r217442 = r217440 * r217417;
        double r217443 = r217441 * r217442;
        double r217444 = r217439 - r217443;
        double r217445 = cbrt(r217444);
        double r217446 = r217438 * r217445;
        double r217447 = sqrt(r217446);
        double r217448 = 5.95390581098912e-86;
        bool r217449 = r217412 <= r217448;
        double r217450 = 2.81700744232938e+107;
        bool r217451 = r217412 <= r217450;
        double r217452 = !r217451;
        bool r217453 = r217449 || r217452;
        double r217454 = r217435 * r217415;
        double r217455 = sqrt(r217454);
        double r217456 = pow(r217420, r217422);
        double r217457 = r217427 * r217456;
        double r217458 = r217457 * r217417;
        double r217459 = r217439 - r217458;
        double r217460 = sqrt(r217459);
        double r217461 = r217455 * r217460;
        double r217462 = r217454 * r217444;
        double r217463 = sqrt(r217462);
        double r217464 = r217453 ? r217461 : r217463;
        double r217465 = r217414 ? r217447 : r217464;
        return r217465;
}

Error

Bits error versus n

Bits error versus U

Bits error versus t

Bits error versus l

Bits error versus Om

Bits error versus U*

Derivation

1. Split input into 3 regimes

2. if t < 3.1306329293196524e-271

   1. Initial program 35.5

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied associate-/l* 32.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 32.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\]

   6. Applied associate-*r* 31.8

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\]
   7. Using strategy rm

   8. Applied associate-*l* 31.9

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}\]
   9. Using strategy rm

   10. Applied add-cube-cbrt 32.2

       \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \color{blue}{\left(\left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}}\]

   11. Applied associate-*r* 32.2

       \[\leadsto \sqrt{\color{blue}{\left(\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}}}\]

   12. Simplified 32.7

       \[\leadsto \sqrt{\color{blue}{\left(\sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} \cdot \left(U \cdot \left(\left(2 \cdot n\right) \cdot \sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\right)\right)} \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}}\]

   if 3.1306329293196524e-271 < t < 5.95390581098912e-86 or 2.81700744232938e+107 < t

   1. Initial program 36.1

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied associate-/l* 33.3

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   4. Using strategy rm

   5. Applied sqrt-prod 29.4

      \[\leadsto \color{blue}{\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}}\]

   if 5.95390581098912e-86 < t < 2.81700744232938e+107

   1. Initial program 31.3

      \[\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell \cdot \ell}{Om}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   2. Using strategy rm

   3. Applied associate-/l* 28.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \color{blue}{\frac{\ell}{\frac{Om}{\ell}}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\]
   4. Using strategy rm

   5. Applied sqr-pow 28.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot \color{blue}{\left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot \left(U - U*\right)\right)}\]

   6. Applied associate-*r* 27.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot \left(U - U*\right)\right)}\]
   7. Using strategy rm

   8. Applied associate-*l* 27.4

      \[\leadsto \sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \color{blue}{\left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}\right)}\]
3. Recombined 3 regimes into one program.

4. Final simplification 30.9

   \[\leadsto \begin{array}{l} \mathbf{if}\;t \le 3.130632929319652416005471644268637375687 \cdot 10^{-271}:\\ \;\;\;\;\sqrt{\left(\sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)} \cdot \left(U \cdot \left(\left(2 \cdot n\right) \cdot \sqrt[3]{t - \left(\left(\left(U - U*\right) \cdot {\left(\frac{\ell}{Om}\right)}^{\left(2 \cdot \frac{2}{2}\right)}\right) \cdot n + 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right)}\right)\right)\right) \cdot \sqrt[3]{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)}}\\ \mathbf{elif}\;t \le 5.953905810989119972468590963716396861846 \cdot 10^{-86} \lor \neg \left(t \le 2.817007442329380152502379998300985537163 \cdot 10^{107}\right):\\ \;\;\;\;\sqrt{\left(2 \cdot n\right) \cdot U} \cdot \sqrt{\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - 2 \cdot \frac{\ell}{\frac{Om}{\ell}}\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot \left({\left(\frac{\ell}{Om}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(U - U*\right)\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019350 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  :precision binary64
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))