Henrywood and Agarwal, Equation (12)

Average Error: 26.0 → 16.4

Time: 1.2m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r5237219 = d;
        double r5237220 = h;
        double r5237221 = r5237219 / r5237220;
        double r5237222 = 1.0;
        double r5237223 = 2.0;
        double r5237224 = r5237222 / r5237223;
        double r5237225 = pow(r5237221, r5237224);
        double r5237226 = l;
        double r5237227 = r5237219 / r5237226;
        double r5237228 = pow(r5237227, r5237224);
        double r5237229 = r5237225 * r5237228;
        double r5237230 = M;
        double r5237231 = D;
        double r5237232 = r5237230 * r5237231;
        double r5237233 = r5237223 * r5237219;
        double r5237234 = r5237232 / r5237233;
        double r5237235 = pow(r5237234, r5237223);
        double r5237236 = r5237224 * r5237235;
        double r5237237 = r5237220 / r5237226;
        double r5237238 = r5237236 * r5237237;
        double r5237239 = r5237222 - r5237238;
        double r5237240 = r5237229 * r5237239;
        return r5237240;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r5237241 = d;
        double r5237242 = cbrt(r5237241);
        double r5237243 = h;
        double r5237244 = r5237242 / r5237243;
        double r5237245 = sqrt(r5237244);
        double r5237246 = fabs(r5237242);
        double r5237247 = r5237245 * r5237246;
        double r5237248 = l;
        double r5237249 = cbrt(r5237248);
        double r5237250 = r5237242 / r5237249;
        double r5237251 = fabs(r5237250);
        double r5237252 = sqrt(r5237250);
        double r5237253 = r5237251 * r5237252;
        double r5237254 = r5237247 * r5237253;
        double r5237255 = -0.5;
        double r5237256 = 1.0;
        double r5237257 = r5237256 / r5237248;
        double r5237258 = M;
        double r5237259 = 2.0;
        double r5237260 = D;
        double r5237261 = r5237260 / r5237241;
        double r5237262 = r5237259 / r5237261;
        double r5237263 = r5237258 / r5237262;
        double r5237264 = r5237263 * r5237243;
        double r5237265 = r5237257 * r5237264;
        double r5237266 = r5237265 * r5237263;
        double r5237267 = r5237255 * r5237266;
        double r5237268 = r5237241 / r5237243;
        double r5237269 = sqrt(r5237268);
        double r5237270 = r5237269 * r5237253;
        double r5237271 = fma(r5237254, r5237267, r5237270);
        return r5237271;
}

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 26.0

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

2. Simplified 25.0

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

4. Applied add-cube-cbrt 25.3

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

5. Applied add-cube-cbrt 25.3

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

6. Applied times-frac 25.3

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

7. Applied sqrt-prod 23.1

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

8. Simplified 22.9

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

10. Applied add-cube-cbrt 22.9

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

11. Applied add-cube-cbrt 22.9

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

12. Applied times-frac 22.9

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

13. Applied sqrt-prod 20.8

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

14. Simplified 20.8

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

16. Applied div-inv 20.8

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

17. Applied associate-*r* 16.9

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

19. Applied *-un-lft-identity 16.9

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

20. Applied add-cube-cbrt 16.9

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

21. Applied times-frac 16.9

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

22. Applied sqrt-prod 16.4

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

23. Simplified 16.4

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

24. Final simplification 16.4

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

Reproduce

herbie shell --seed 2019138 +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)))))