Henrywood and Agarwal, Equation (12)

Average Error: 24.9 → 17.5

Time: 1.0m

Precision: 64

Math

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

↓

\[\mathsf{fma}\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{d}{h}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\frac{\sqrt[3]{h}}{\ell} \cdot \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)\right)\right) \cdot \frac{-1}{2}, \left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)\right)\]

C

double f(double d, double h, double l, double M, double D) {
        double r5198259 = d;
        double r5198260 = h;
        double r5198261 = r5198259 / r5198260;
        double r5198262 = 1.0;
        double r5198263 = 2.0;
        double r5198264 = r5198262 / r5198263;
        double r5198265 = pow(r5198261, r5198264);
        double r5198266 = l;
        double r5198267 = r5198259 / r5198266;
        double r5198268 = pow(r5198267, r5198264);
        double r5198269 = r5198265 * r5198268;
        double r5198270 = M;
        double r5198271 = D;
        double r5198272 = r5198270 * r5198271;
        double r5198273 = r5198263 * r5198259;
        double r5198274 = r5198272 / r5198273;
        double r5198275 = pow(r5198274, r5198263);
        double r5198276 = r5198264 * r5198275;
        double r5198277 = r5198260 / r5198266;
        double r5198278 = r5198276 * r5198277;
        double r5198279 = r5198262 - r5198278;
        double r5198280 = r5198269 * r5198279;
        return r5198280;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5198281 = d;
        double r5198282 = cbrt(r5198281);
        double r5198283 = l;
        double r5198284 = cbrt(r5198283);
        double r5198285 = r5198282 / r5198284;
        double r5198286 = fabs(r5198285);
        double r5198287 = sqrt(r5198285);
        double r5198288 = r5198286 * r5198287;
        double r5198289 = h;
        double r5198290 = r5198281 / r5198289;
        double r5198291 = sqrt(r5198290);
        double r5198292 = r5198288 * r5198291;
        double r5198293 = M;
        double r5198294 = 2.0;
        double r5198295 = r5198294 * r5198281;
        double r5198296 = D;
        double r5198297 = r5198295 / r5198296;
        double r5198298 = r5198293 / r5198297;
        double r5198299 = cbrt(r5198289);
        double r5198300 = r5198299 / r5198283;
        double r5198301 = r5198299 * r5198299;
        double r5198302 = r5198298 * r5198301;
        double r5198303 = r5198300 * r5198302;
        double r5198304 = r5198298 * r5198303;
        double r5198305 = -0.5;
        double r5198306 = r5198304 * r5198305;
        double r5198307 = 1.0;
        double r5198308 = r5198307 / r5198301;
        double r5198309 = sqrt(r5198308);
        double r5198310 = r5198281 / r5198299;
        double r5198311 = sqrt(r5198310);
        double r5198312 = r5198309 * r5198311;
        double r5198313 = r5198281 / r5198284;
        double r5198314 = sqrt(r5198313);
        double r5198315 = r5198284 * r5198284;
        double r5198316 = r5198307 / r5198315;
        double r5198317 = sqrt(r5198316);
        double r5198318 = r5198314 * r5198317;
        double r5198319 = r5198312 * r5198318;
        double r5198320 = fma(r5198292, r5198306, r5198319);
        return r5198320;
}

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

1. Initial program 24.9

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

2. Simplified 24.0

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

4. Applied *-un-lft-identity 24.0

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

5. Applied add-cube-cbrt 24.1

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

6. Applied times-frac 24.1

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

7. Applied associate-*r* 22.4

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

9. Applied add-cube-cbrt 22.6

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

10. Applied *-un-lft-identity 22.6

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

11. Applied times-frac 22.6

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

12. Applied sqrt-prod 20.4

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

14. Applied add-cube-cbrt 20.4

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

15. Applied add-cube-cbrt 20.4

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

16. Applied times-frac 20.4

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

17. Applied sqrt-prod 17.8

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

18. Simplified 17.8

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

20. Applied add-cube-cbrt 17.9

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

21. Applied *-un-lft-identity 17.9

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

22. Applied times-frac 17.9

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

23. Applied sqrt-prod 17.5

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

24. Final simplification 17.5

    \[\leadsto \mathsf{fma}\left(\left(\left|\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}\right| \cdot \sqrt{\frac{\sqrt[3]{d}}{\sqrt[3]{\ell}}}\right) \cdot \sqrt{\frac{d}{h}}, \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\frac{\sqrt[3]{h}}{\ell} \cdot \left(\frac{M}{\frac{2 \cdot d}{D}} \cdot \left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right)\right)\right)\right) \cdot \frac{-1}{2}, \left(\sqrt{\frac{1}{\sqrt[3]{h} \cdot \sqrt[3]{h}}} \cdot \sqrt{\frac{d}{\sqrt[3]{h}}}\right) \cdot \left(\sqrt{\frac{d}{\sqrt[3]{\ell}}} \cdot \sqrt{\frac{1}{\sqrt[3]{\ell} \cdot \sqrt[3]{\ell}}}\right)\right)\]

Reproduce

herbie shell --seed 2019158 +o rules:numerics
(FPCore (d h l M D)
  :name "Henrywood and Agarwal, Equation (12)"
  (* (* (pow (/ d h) (/ 1 2)) (pow (/ d l) (/ 1 2))) (- 1 (* (* (/ 1 2) (pow (/ (* M D) (* 2 d)) 2)) (/ h l)))))