Average Error: 26.1 → 18.5
Time: 1.5m
Precision: 64
Internal Precision: 128
\[\left({\left(\frac{d}{h}\right)}^{\left(\frac{1}{2}\right)} \cdot {\left(\frac{d}{\ell}\right)}^{\left(\frac{1}{2}\right)}\right) \cdot \left(1 - \left(\frac{1}{2} \cdot {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2}\right) \cdot \frac{h}{\ell}\right)\]
↓
\[\begin{array}{l} \mathbf{if}\;d \le 2.94647775810176 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\sqrt[3]{\frac{d}{h}}} \cdot \left|\sqrt[3]{\frac{d}{h}}\right|}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot \ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}\right)\\ \mathbf{elif}\;d \le 1.2261136402612018 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;d \le 2.2562475365295626 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\sqrt[3]{\frac{d}{h}}} \cdot \left|\sqrt[3]{\frac{d}{h}}\right|}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot \ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Error

Bits error versus d

Bits error versus h

Bits error versus l

Bits error versus M

Bits error versus D

Derivation

  1. Split input into 2 regimes

  2. if d < 2.94647775810176e-310 or 1.2261136402612018e-70 < d < 2.2562475365295626e+138

    1. Initial program 23.5

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

    2. Simplified23.2

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

    4. Applied associate-/l*21.8

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

    6. Applied add-sqr-sqrt21.8

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

    7. Applied sqrt-prod22.0

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

    9. Applied add-cube-cbrt22.0

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

    10. Applied sqrt-prod22.0

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

    11. Simplified22.0

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

    13. Applied associate-*l/22.0

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

    14. Applied associate-/l/19.8

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

    if 2.94647775810176e-310 < d < 1.2261136402612018e-70 or 2.2562475365295626e+138 < d

    1. Initial program 31.9

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

    2. Simplified31.7

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

    4. Applied associate-/l*30.3

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

    6. Applied sqrt-div23.0

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

    7. Applied sqrt-div16.6

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

    8. Applied frac-times16.6

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

    9. Applied associate-*r/15.5

      \[\leadsto \color{blue}{\frac{\left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right) \cdot \left(\sqrt{d} \cdot \sqrt{d}\right)}{\sqrt{\ell} \cdot \sqrt{h}}}\]
  3. Recombined 2 regimes into one program.

  4. Final simplification18.5

    \[\leadsto \begin{array}{l} \mathbf{if}\;d \le 2.94647775810176 \cdot 10^{-310}:\\ \;\;\;\;\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\sqrt[3]{\frac{d}{h}}} \cdot \left|\sqrt[3]{\frac{d}{h}}\right|}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot \ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}\right)\\ \mathbf{elif}\;d \le 1.2261136402612018 \cdot 10^{-70}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \mathbf{elif}\;d \le 2.2562475365295626 \cdot 10^{+138}:\\ \;\;\;\;\left(\left(\sqrt{\sqrt{\frac{d}{h}}} \cdot \sqrt{\sqrt{\sqrt[3]{\frac{d}{h}}} \cdot \left|\sqrt[3]{\frac{d}{h}}\right|}\right) \cdot \sqrt{\frac{d}{\ell}}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{2 \cdot \ell}{\frac{M \cdot D}{2 \cdot d} \cdot h}}\right)\\ \mathbf{else}:\\ \;\;\;\;\frac{\left(\sqrt{d} \cdot \sqrt{d}\right) \cdot \left(1 - \frac{\frac{M \cdot D}{2 \cdot d}}{\frac{\frac{\ell}{h} \cdot 2}{\frac{M \cdot D}{2 \cdot d}}}\right)}{\sqrt{\ell} \cdot \sqrt{h}}\\ \end{array}\]

Reproduce

herbie shell --seed 2019050 
(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)))))