Henrywood and Agarwal, Equation (9a)

Average Error: 14.3 → 9.3

Time: 40.9s

Precision: 64

Math

\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

↓

\[\begin{array}{l} \mathbf{if}\;h \le -8.06126815033417573653781222780111910642 \cdot 10^{-242} \lor \neg \left(h \le 131425282164006355827789572611354426081300\right):\\ \;\;\;\;\sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left({\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\\ \end{array}\]

C

double f(double w0, double M, double D, double h, double l, double d) {
        double r166372 = w0;
        double r166373 = 1.0;
        double r166374 = M;
        double r166375 = D;
        double r166376 = r166374 * r166375;
        double r166377 = 2.0;
        double r166378 = d;
        double r166379 = r166377 * r166378;
        double r166380 = r166376 / r166379;
        double r166381 = pow(r166380, r166377);
        double r166382 = h;
        double r166383 = l;
        double r166384 = r166382 / r166383;
        double r166385 = r166381 * r166384;
        double r166386 = r166373 - r166385;
        double r166387 = sqrt(r166386);
        double r166388 = r166372 * r166387;
        return r166388;
}

↓

double f(double w0, double M, double D, double h, double l, double d) {
        double r166389 = h;
        double r166390 = -8.061268150334176e-242;
        bool r166391 = r166389 <= r166390;
        double r166392 = 1.3142528216400636e+41;
        bool r166393 = r166389 <= r166392;
        double r166394 = !r166393;
        bool r166395 = r166391 || r166394;
        double r166396 = 1.0;
        double r166397 = M;
        double r166398 = 2.0;
        double r166399 = r166397 / r166398;
        double r166400 = D;
        double r166401 = d;
        double r166402 = r166400 / r166401;
        double r166403 = r166399 * r166402;
        double r166404 = 2.0;
        double r166405 = r166398 / r166404;
        double r166406 = pow(r166403, r166405);
        double r166407 = l;
        double r166408 = cbrt(r166407);
        double r166409 = r166406 / r166408;
        double r166410 = 1.0;
        double r166411 = sqrt(r166398);
        double r166412 = r166410 / r166411;
        double r166413 = pow(r166412, r166405);
        double r166414 = r166389 * r166413;
        double r166415 = r166408 * r166408;
        double r166416 = r166397 / r166411;
        double r166417 = pow(r166416, r166405);
        double r166418 = r166415 / r166417;
        double r166419 = pow(r166402, r166405);
        double r166420 = r166418 / r166419;
        double r166421 = r166414 / r166420;
        double r166422 = r166409 * r166421;
        double r166423 = r166396 - r166422;
        double r166424 = sqrt(r166423);
        double r166425 = w0;
        double r166426 = r166424 * r166425;
        double r166427 = r166411 / r166400;
        double r166428 = r166397 / r166427;
        double r166429 = r166428 / r166401;
        double r166430 = pow(r166429, r166405);
        double r166431 = r166430 * r166414;
        double r166432 = r166397 * r166400;
        double r166433 = r166398 * r166401;
        double r166434 = r166432 / r166433;
        double r166435 = pow(r166434, r166405);
        double r166436 = r166431 * r166435;
        double r166437 = r166436 / r166407;
        double r166438 = r166396 - r166437;
        double r166439 = sqrt(r166438);
        double r166440 = r166439 * r166425;
        double r166441 = r166395 ? r166426 : r166440;
        return r166441;
}

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

1. Split input into 2 regimes

2. if h < -8.061268150334176e-242 or 1.3142528216400636e+41 < h

   1. Initial program 16.5

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 11.6

      \[\leadsto \color{blue}{\sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}} \cdot w0}\]
   3. Using strategy rm

   4. Applied sqr-pow 11.6

      \[\leadsto \sqrt{1 - \frac{h \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}}{\ell}} \cdot w0\]

   5. Applied associate-*r* 10.7

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{\ell}} \cdot w0\]

   6. Simplified 10.7

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 10.7

      \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{D}}}{\color{blue}{1 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   9. Applied *-un-lft-identity 10.7

      \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{\color{blue}{1 \cdot D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   10. Applied add-sqr-sqrt 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot D}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   11. Applied times-frac 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   12. Applied *-un-lft-identity 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{\color{blue}{1 \cdot M}}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{D}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   13. Applied times-frac 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\color{blue}{\frac{1}{\frac{\sqrt{2}}{1}} \cdot \frac{M}{\frac{\sqrt{2}}{D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   14. Applied times-frac 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\color{blue}{\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1} \cdot \frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   15. Applied unpow-prod-down 10.7

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot \color{blue}{\left({\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   16. Applied associate-*r* 10.7

       \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(\left(h \cdot {\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   17. Simplified 10.7

       \[\leadsto \sqrt{1 - \frac{\left(\color{blue}{\left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]
   18. Using strategy rm

   19. Applied add-cube-cbrt 10.7

       \[\leadsto \sqrt{1 - \frac{\left(\left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\color{blue}{\left(\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}\right) \cdot \sqrt[3]{\ell}}}} \cdot w0\]

   20. Applied times-frac 9.7

       \[\leadsto \sqrt{1 - \color{blue}{\frac{\left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}} \cdot w0\]

   21. Simplified 10.2

       \[\leadsto \sqrt{1 - \color{blue}{\frac{{\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\frac{M}{\sqrt{2}} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot \frac{{\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}} \cdot w0\]

   22. Simplified 9.2

       \[\leadsto \sqrt{1 - \frac{{\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\frac{M}{\sqrt{2}} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}} \cdot \color{blue}{\frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}}} \cdot w0\]
   23. Using strategy rm

   24. Applied unpow-prod-down 9.2

       \[\leadsto \sqrt{1 - \frac{{\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{\color{blue}{{\left(\frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}} \cdot w0\]

   25. Applied associate-/r* 10.1

       \[\leadsto \sqrt{1 - \frac{{\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)} \cdot h}{\color{blue}{\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}}} \cdot w0\]

   if -8.061268150334176e-242 < h < 1.3142528216400636e+41

   1. Initial program 10.7

      \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

   2. Simplified 10.6

      \[\leadsto \color{blue}{\sqrt{1 - \frac{h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}}{\ell}} \cdot w0}\]
   3. Using strategy rm

   4. Applied sqr-pow 10.6

      \[\leadsto \sqrt{1 - \frac{h \cdot \color{blue}{\left({\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right)}}{\ell}} \cdot w0\]

   5. Applied associate-*r* 8.0

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}}{\ell}} \cdot w0\]

   6. Simplified 8.0

      \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]
   7. Using strategy rm

   8. Applied *-un-lft-identity 8.0

      \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{D}}}{\color{blue}{1 \cdot d}}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   9. Applied *-un-lft-identity 8.0

      \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{2}{\color{blue}{1 \cdot D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   10. Applied add-sqr-sqrt 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\frac{\color{blue}{\sqrt{2} \cdot \sqrt{2}}}{1 \cdot D}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   11. Applied times-frac 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{M}{\color{blue}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   12. Applied *-un-lft-identity 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\frac{\color{blue}{1 \cdot M}}{\frac{\sqrt{2}}{1} \cdot \frac{\sqrt{2}}{D}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   13. Applied times-frac 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\left(\frac{\color{blue}{\frac{1}{\frac{\sqrt{2}}{1}} \cdot \frac{M}{\frac{\sqrt{2}}{D}}}}{1 \cdot d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   14. Applied times-frac 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot {\color{blue}{\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1} \cdot \frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   15. Applied unpow-prod-down 8.0

       \[\leadsto \sqrt{1 - \frac{\left(h \cdot \color{blue}{\left({\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1}\right)}^{\left(\frac{2}{2}\right)} \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   16. Applied associate-*r* 8.0

       \[\leadsto \sqrt{1 - \frac{\color{blue}{\left(\left(h \cdot {\left(\frac{\frac{1}{\frac{\sqrt{2}}{1}}}{1}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]

   17. Simplified 8.0

       \[\leadsto \sqrt{1 - \frac{\left(\color{blue}{\left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right)} \cdot {\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)}\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\]
3. Recombined 2 regimes into one program.

4. Final simplification 9.3

   \[\leadsto \begin{array}{l} \mathbf{if}\;h \le -8.06126815033417573653781222780111910642 \cdot 10^{-242} \lor \neg \left(h \le 131425282164006355827789572611354426081300\right):\\ \;\;\;\;\sqrt{1 - \frac{{\left(\frac{M}{2} \cdot \frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}{\sqrt[3]{\ell}} \cdot \frac{h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}{\frac{\frac{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}{{\left(\frac{M}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}}}{{\left(\frac{D}{d}\right)}^{\left(\frac{2}{2}\right)}}}} \cdot w0\\ \mathbf{else}:\\ \;\;\;\;\sqrt{1 - \frac{\left({\left(\frac{\frac{M}{\frac{\sqrt{2}}{D}}}{d}\right)}^{\left(\frac{2}{2}\right)} \cdot \left(h \cdot {\left(\frac{1}{\sqrt{2}}\right)}^{\left(\frac{2}{2}\right)}\right)\right) \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{\left(\frac{2}{2}\right)}}{\ell}} \cdot w0\\ \end{array}\]

Reproduce

herbie shell --seed 2019174 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1.0 (* (pow (/ (* M D) (* 2.0 d)) 2.0) (/ h l))))))