Average Error: 13.5 → 9.2
Time: 5.3m
Precision: 64
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot {\left(1 - \left(\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \sqrt[3]{h}}{2 \cdot d}\right) \cdot \frac{\left(D \cdot M\right) \cdot \sqrt[3]{h}}{2 \cdot d}\right)}^{\frac{1}{2}}\]

Error

Bits error versus w0

Bits error versus M

Bits error versus D

Bits error versus h

Bits error versus l

Bits error versus d

Derivation

  1. Initial program 13.5

    \[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]

  2. Simplified13.5

    \[\leadsto \color{blue}{\sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\ell}} \cdot w0}\]
  3. Using strategy rm

  4. Applied *-un-lft-identity13.5

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{h}{\color{blue}{1 \cdot \ell}}} \cdot w0\]

  5. Applied add-cube-cbrt13.6

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\color{blue}{\left(\sqrt[3]{h} \cdot \sqrt[3]{h}\right) \cdot \sqrt[3]{h}}}{1 \cdot \ell}} \cdot w0\]

  6. Applied times-frac13.6

    \[\leadsto \sqrt{1 - \left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \color{blue}{\left(\frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1} \cdot \frac{\sqrt[3]{h}}{\ell}\right)}} \cdot w0\]

  7. Applied associate-*r*11.3

    \[\leadsto \sqrt{1 - \color{blue}{\left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}}} \cdot w0\]
  8. Using strategy rm

  9. Applied pow111.3

    \[\leadsto \sqrt{\color{blue}{{\left(1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}^{1}}} \cdot w0\]

  10. Applied sqrt-pow111.3

    \[\leadsto \color{blue}{{\left(1 - \left(\left(\frac{M \cdot D}{2 \cdot d} \cdot \frac{M \cdot D}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \sqrt[3]{h}}{1}\right) \cdot \frac{\sqrt[3]{h}}{\ell}\right)}^{\left(\frac{1}{2}\right)}} \cdot w0\]

  11. Simplified9.2

    \[\leadsto {\color{blue}{\left(1 - \left(\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\sqrt[3]{h} \cdot \left(D \cdot M\right)}{2 \cdot d}\right) \cdot \frac{\sqrt[3]{h} \cdot \left(D \cdot M\right)}{2 \cdot d}\right)}}^{\left(\frac{1}{2}\right)} \cdot w0\]

  12. Final simplification9.2

    \[\leadsto w0 \cdot {\left(1 - \left(\frac{\sqrt[3]{h}}{\ell} \cdot \frac{\left(D \cdot M\right) \cdot \sqrt[3]{h}}{2 \cdot d}\right) \cdot \frac{\left(D \cdot M\right) \cdot \sqrt[3]{h}}{2 \cdot d}\right)}^{\frac{1}{2}}\]

Reproduce

herbie shell --seed 2019100 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))