Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 17.7

Time: 1.0m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r6795371 = d;
        double r6795372 = h;
        double r6795373 = r6795371 / r6795372;
        double r6795374 = 1.0;
        double r6795375 = 2.0;
        double r6795376 = r6795374 / r6795375;
        double r6795377 = pow(r6795373, r6795376);
        double r6795378 = l;
        double r6795379 = r6795371 / r6795378;
        double r6795380 = pow(r6795379, r6795376);
        double r6795381 = r6795377 * r6795380;
        double r6795382 = M;
        double r6795383 = D;
        double r6795384 = r6795382 * r6795383;
        double r6795385 = r6795375 * r6795371;
        double r6795386 = r6795384 / r6795385;
        double r6795387 = pow(r6795386, r6795375);
        double r6795388 = r6795376 * r6795387;
        double r6795389 = r6795372 / r6795378;
        double r6795390 = r6795388 * r6795389;
        double r6795391 = r6795374 - r6795390;
        double r6795392 = r6795381 * r6795391;
        return r6795392;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r6795393 = d;
        double r6795394 = cbrt(r6795393);
        double r6795395 = l;
        double r6795396 = cbrt(r6795395);
        double r6795397 = r6795394 / r6795396;
        double r6795398 = fabs(r6795397);
        double r6795399 = sqrt(r6795397);
        double r6795400 = r6795398 * r6795399;
        double r6795401 = h;
        double r6795402 = r6795393 / r6795401;
        double r6795403 = sqrt(r6795402);
        double r6795404 = r6795400 * r6795403;
        double r6795405 = M;
        double r6795406 = 2.0;
        double r6795407 = r6795406 * r6795393;
        double r6795408 = D;
        double r6795409 = r6795407 / r6795408;
        double r6795410 = r6795405 / r6795409;
        double r6795411 = cbrt(r6795401);
        double r6795412 = r6795411 / r6795395;
        double r6795413 = r6795411 * r6795411;
        double r6795414 = r6795410 * r6795413;
        double r6795415 = r6795412 * r6795414;
        double r6795416 = r6795410 * r6795415;
        double r6795417 = -0.5;
        double r6795418 = r6795416 * r6795417;
        double r6795419 = r6795393 / r6795396;
        double r6795420 = sqrt(r6795419);
        double r6795421 = 1.0;
        double r6795422 = r6795396 * r6795396;
        double r6795423 = r6795421 / r6795422;
        double r6795424 = sqrt(r6795423);
        double r6795425 = r6795420 * r6795424;
        double r6795426 = r6795394 / r6795411;
        double r6795427 = fabs(r6795426);
        double r6795428 = sqrt(r6795426);
        double r6795429 = r6795427 * r6795428;
        double r6795430 = r6795425 * r6795429;
        double r6795431 = fma(r6795404, r6795418, r6795430);
        return r6795431;
}

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 25.6

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

2. Simplified 24.5

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

4. Applied add-cube-cbrt 24.7

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

5. Applied *-un-lft-identity 24.7

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

6. Applied times-frac 24.7

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

7. Applied sqrt-prod 22.5

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

9. Applied add-cube-cbrt 22.5

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

10. Applied add-cube-cbrt 22.6

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

11. Applied times-frac 22.6

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

12. Applied sqrt-prod 20.7

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

13. Simplified 20.7

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

15. Applied *-un-lft-identity 20.7

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

16. Applied add-cube-cbrt 20.8

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

17. Applied times-frac 20.8

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

18. Applied associate-*r* 17.9

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

20. Applied add-cube-cbrt 18.0

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

21. Applied add-cube-cbrt 18.1

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

22. Applied times-frac 18.1

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

23. Applied sqrt-prod 17.7

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

24. Simplified 17.7

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

25. Final simplification 17.7

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

Reproduce

herbie shell --seed 2019162 +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)))))