Henrywood and Agarwal, Equation (12)

Average Error: 25.6 → 17.7

Time: 51.5s

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 r6973094 = d;
        double r6973095 = h;
        double r6973096 = r6973094 / r6973095;
        double r6973097 = 1.0;
        double r6973098 = 2.0;
        double r6973099 = r6973097 / r6973098;
        double r6973100 = pow(r6973096, r6973099);
        double r6973101 = l;
        double r6973102 = r6973094 / r6973101;
        double r6973103 = pow(r6973102, r6973099);
        double r6973104 = r6973100 * r6973103;
        double r6973105 = M;
        double r6973106 = D;
        double r6973107 = r6973105 * r6973106;
        double r6973108 = r6973098 * r6973094;
        double r6973109 = r6973107 / r6973108;
        double r6973110 = pow(r6973109, r6973098);
        double r6973111 = r6973099 * r6973110;
        double r6973112 = r6973095 / r6973101;
        double r6973113 = r6973111 * r6973112;
        double r6973114 = r6973097 - r6973113;
        double r6973115 = r6973104 * r6973114;
        return r6973115;
}

↓

double f(double d, double h, double l, double M, double D) {
        double r6973116 = d;
        double r6973117 = cbrt(r6973116);
        double r6973118 = l;
        double r6973119 = cbrt(r6973118);
        double r6973120 = r6973117 / r6973119;
        double r6973121 = fabs(r6973120);
        double r6973122 = sqrt(r6973120);
        double r6973123 = r6973121 * r6973122;
        double r6973124 = h;
        double r6973125 = r6973116 / r6973124;
        double r6973126 = sqrt(r6973125);
        double r6973127 = r6973123 * r6973126;
        double r6973128 = M;
        double r6973129 = 2.0;
        double r6973130 = r6973129 * r6973116;
        double r6973131 = D;
        double r6973132 = r6973130 / r6973131;
        double r6973133 = r6973128 / r6973132;
        double r6973134 = cbrt(r6973124);
        double r6973135 = r6973134 / r6973118;
        double r6973136 = r6973134 * r6973134;
        double r6973137 = r6973133 * r6973136;
        double r6973138 = r6973135 * r6973137;
        double r6973139 = r6973133 * r6973138;
        double r6973140 = -0.5;
        double r6973141 = r6973139 * r6973140;
        double r6973142 = r6973116 / r6973119;
        double r6973143 = sqrt(r6973142);
        double r6973144 = 1.0;
        double r6973145 = r6973119 * r6973119;
        double r6973146 = r6973144 / r6973145;
        double r6973147 = sqrt(r6973146);
        double r6973148 = r6973143 * r6973147;
        double r6973149 = r6973117 / r6973134;
        double r6973150 = fabs(r6973149);
        double r6973151 = sqrt(r6973149);
        double r6973152 = r6973150 * r6973151;
        double r6973153 = r6973148 * r6973152;
        double r6973154 = fma(r6973127, r6973141, r6973153);
        return r6973154;
}

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)))))