Henrywood and Agarwal, Equation (12)

Average Error: 25.8 → 17.7

Time: 1.6m

Precision: 64

Math

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

↓

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

C

double f(double d, double h, double l, double M, double D) {
        double r8557558 = d;
        double r8557559 = h;
        double r8557560 = r8557558 / r8557559;
        double r8557561 = 1.0;
        double r8557562 = 2.0;
        double r8557563 = r8557561 / r8557562;
        double r8557564 = pow(r8557560, r8557563);
        double r8557565 = l;
        double r8557566 = r8557558 / r8557565;
        double r8557567 = pow(r8557566, r8557563);
        double r8557568 = r8557564 * r8557567;
        double r8557569 = M;
        double r8557570 = D;
        double r8557571 = r8557569 * r8557570;
        double r8557572 = r8557562 * r8557558;
        double r8557573 = r8557571 / r8557572;
        double r8557574 = pow(r8557573, r8557562);
        double r8557575 = r8557563 * r8557574;
        double r8557576 = r8557559 / r8557565;
        double r8557577 = r8557575 * r8557576;
        double r8557578 = r8557561 - r8557577;
        double r8557579 = r8557568 * r8557578;
        return r8557579;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r8557580 = 1.0;
        double r8557581 = h;
        double r8557582 = cbrt(r8557581);
        double r8557583 = r8557582 * r8557582;
        double r8557584 = r8557580 / r8557583;
        double r8557585 = sqrt(r8557584);
        double r8557586 = d;
        double r8557587 = r8557586 / r8557582;
        double r8557588 = sqrt(r8557587);
        double r8557589 = r8557585 * r8557588;
        double r8557590 = cbrt(r8557586);
        double r8557591 = fabs(r8557590);
        double r8557592 = l;
        double r8557593 = r8557590 / r8557592;
        double r8557594 = sqrt(r8557593);
        double r8557595 = r8557591 * r8557594;
        double r8557596 = r8557589 * r8557595;
        double r8557597 = M;
        double r8557598 = 2.0;
        double r8557599 = D;
        double r8557600 = r8557599 / r8557586;
        double r8557601 = r8557598 / r8557600;
        double r8557602 = r8557597 / r8557601;
        double r8557603 = cbrt(r8557592);
        double r8557604 = r8557603 * r8557603;
        double r8557605 = r8557604 / r8557583;
        double r8557606 = r8557602 / r8557605;
        double r8557607 = -0.5;
        double r8557608 = r8557607 * r8557602;
        double r8557609 = r8557603 / r8557582;
        double r8557610 = r8557608 / r8557609;
        double r8557611 = r8557606 * r8557610;
        double r8557612 = r8557586 / r8557581;
        double r8557613 = sqrt(r8557612);
        double r8557614 = r8557613 * r8557595;
        double r8557615 = fma(r8557596, r8557611, r8557614);
        return r8557615;
}

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 25.8

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

2. Simplified 26.0

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

4. Applied add-cube-cbrt 26.1

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

5. Applied add-cube-cbrt 26.1

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

6. Applied times-frac 26.1

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

7. Applied times-frac 23.1

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

9. Applied *-un-lft-identity 23.1

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

10. Applied add-cube-cbrt 23.3

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

11. Applied times-frac 23.3

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

12. Applied sqrt-prod 21.5

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

13. Simplified 21.5

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

15. Applied *-un-lft-identity 21.5

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

16. Applied add-cube-cbrt 21.5

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

17. Applied times-frac 21.5

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

18. Applied sqrt-prod 18.4

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

19. Simplified 18.4

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

21. Applied add-cube-cbrt 18.3

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

22. Applied *-un-lft-identity 18.3

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

23. Applied times-frac 18.3

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

24. Applied sqrt-prod 17.7

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

25. Final simplification 17.7

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

Reproduce

herbie shell --seed 2019133 +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)))))