Toniolo and Linder, Equation (13)

Average Error: 32.8 → 23.3

Time: 37.2s

Precision: 64

Math

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

↓

\[\begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left(\left(2 \cdot \ell - \left(\left(U* - U\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 9.477739406878279 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{\ell}{Om} \cdot \left(-n\right)\right) \cdot \left(U \cdot \left(2 \cdot \ell - \left(\left(U* - U\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right) + U \cdot \left(t \cdot n\right)\right)}\\ \end{array}\]

C

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2976329 = 2.0;
        double r2976330 = n;
        double r2976331 = r2976329 * r2976330;
        double r2976332 = U;
        double r2976333 = r2976331 * r2976332;
        double r2976334 = t;
        double r2976335 = l;
        double r2976336 = r2976335 * r2976335;
        double r2976337 = Om;
        double r2976338 = r2976336 / r2976337;
        double r2976339 = r2976329 * r2976338;
        double r2976340 = r2976334 - r2976339;
        double r2976341 = r2976335 / r2976337;
        double r2976342 = pow(r2976341, r2976329);
        double r2976343 = r2976330 * r2976342;
        double r2976344 = U_;
        double r2976345 = r2976332 - r2976344;
        double r2976346 = r2976343 * r2976345;
        double r2976347 = r2976340 - r2976346;
        double r2976348 = r2976333 * r2976347;
        double r2976349 = sqrt(r2976348);
        return r2976349;
}

↓

double f(double n, double U, double t, double l, double Om, double U_) {
        double r2976350 = 2.0;
        double r2976351 = n;
        double r2976352 = r2976350 * r2976351;
        double r2976353 = U;
        double r2976354 = r2976352 * r2976353;
        double r2976355 = t;
        double r2976356 = l;
        double r2976357 = r2976356 * r2976356;
        double r2976358 = Om;
        double r2976359 = r2976357 / r2976358;
        double r2976360 = r2976359 * r2976350;
        double r2976361 = r2976355 - r2976360;
        double r2976362 = r2976356 / r2976358;
        double r2976363 = pow(r2976362, r2976350);
        double r2976364 = r2976351 * r2976363;
        double r2976365 = U_;
        double r2976366 = r2976353 - r2976365;
        double r2976367 = r2976364 * r2976366;
        double r2976368 = r2976361 - r2976367;
        double r2976369 = r2976354 * r2976368;
        double r2976370 = 0.0;
        bool r2976371 = r2976369 <= r2976370;
        double r2976372 = r2976353 * r2976355;
        double r2976373 = r2976351 * r2976372;
        double r2976374 = r2976350 * r2976356;
        double r2976375 = r2976365 - r2976353;
        double r2976376 = r2976375 * r2976351;
        double r2976377 = r2976376 * r2976362;
        double r2976378 = r2976374 - r2976377;
        double r2976379 = r2976351 * r2976362;
        double r2976380 = r2976378 * r2976379;
        double r2976381 = -r2976353;
        double r2976382 = r2976380 * r2976381;
        double r2976383 = r2976373 + r2976382;
        double r2976384 = r2976350 * r2976383;
        double r2976385 = sqrt(r2976384);
        double r2976386 = 9.477739406878279e+303;
        bool r2976387 = r2976369 <= r2976386;
        double r2976388 = sqrt(r2976369);
        double r2976389 = -r2976351;
        double r2976390 = r2976362 * r2976389;
        double r2976391 = r2976353 * r2976378;
        double r2976392 = r2976390 * r2976391;
        double r2976393 = r2976355 * r2976351;
        double r2976394 = r2976353 * r2976393;
        double r2976395 = r2976392 + r2976394;
        double r2976396 = r2976350 * r2976395;
        double r2976397 = sqrt(r2976396);
        double r2976398 = r2976387 ? r2976388 : r2976397;
        double r2976399 = r2976371 ? r2976385 : r2976398;
        return r2976399;
}

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 (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 0.0

   1. Initial program 56.9

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

   2. Simplified 38.2

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

   4. Applied sub-neg 38.2

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

   5. Applied distribute-lft-in 38.2

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

   7. Applied distribute-rgt-neg-in 38.2

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

   8. Applied associate-*r* 38.2

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

   10. Applied distribute-rgt-in 38.2

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

   12. Applied associate-*l* 38.7

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

   if 0.0 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*)))) < 9.477739406878279e+303

   1. Initial program 1.7

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

   if 9.477739406878279e+303 < (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))

   1. Initial program 60.3

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

   2. Simplified 52.5

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

   4. Applied sub-neg 52.5

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

   5. Applied distribute-lft-in 52.5

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

   7. Applied distribute-rgt-neg-in 52.5

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

   8. Applied associate-*r* 45.4

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

   10. Applied distribute-rgt-in 45.4

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

   12. Applied associate-*l* 43.1

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

4. Final simplification 23.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 0.0:\\ \;\;\;\;\sqrt{2 \cdot \left(n \cdot \left(U \cdot t\right) + \left(\left(2 \cdot \ell - \left(\left(U* - U\right) \cdot n\right) \cdot \frac{\ell}{Om}\right) \cdot \left(n \cdot \frac{\ell}{Om}\right)\right) \cdot \left(-U\right)\right)}\\ \mathbf{elif}\;\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right) \le 9.477739406878279 \cdot 10^{+303}:\\ \;\;\;\;\sqrt{\left(\left(2 \cdot n\right) \cdot U\right) \cdot \left(\left(t - \frac{\ell \cdot \ell}{Om} \cdot 2\right) - \left(n \cdot {\left(\frac{\ell}{Om}\right)}^{2}\right) \cdot \left(U - U*\right)\right)}\\ \mathbf{else}:\\ \;\;\;\;\sqrt{2 \cdot \left(\left(\frac{\ell}{Om} \cdot \left(-n\right)\right) \cdot \left(U \cdot \left(2 \cdot \ell - \left(\left(U* - U\right) \cdot n\right) \cdot \frac{\ell}{Om}\right)\right) + U \cdot \left(t \cdot n\right)\right)}\\ \end{array}\]

Reproduce

herbie shell --seed 2019162 
(FPCore (n U t l Om U*)
  :name "Toniolo and Linder, Equation (13)"
  (sqrt (* (* (* 2 n) U) (- (- t (* 2 (/ (* l l) Om))) (* (* n (pow (/ l Om) 2)) (- U U*))))))