Average Error: 13.5 → 8.7
Time: 1.3m
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\left(\sqrt{\sqrt[3]{1 - \frac{\frac{D \cdot M}{2 \cdot d}}{\ell} \cdot \left(\frac{\sqrt[3]{\frac{1}{\frac{2 \cdot d}{D \cdot M}}} \cdot \sqrt[3]{\frac{1}{\frac{2 \cdot d}{D \cdot M}}}}{\sqrt[3]{\frac{1}{h}} \cdot \sqrt[3]{\frac{1}{h}}} \cdot \frac{\sqrt[3]{\frac{1}{\frac{2 \cdot d}{D \cdot M}}}}{\sqrt[3]{\frac{1}{h}}}\right)}} \cdot \sqrt{\sqrt[3]{1 - \frac{\frac{1}{\frac{2 \cdot d}{D \cdot M}}}{\frac{1}{h}} \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\ell}} \cdot \sqrt[3]{1 - \frac{\frac{1}{\frac{2 \cdot d}{D \cdot M}}}{\frac{1}{h}} \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\ell}}}\right) \cdot w0\]

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. Simplified13.0

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

  4. Applied div-inv13.0

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

  5. Applied times-frac8.6

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

  7. Applied clear-num8.6

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

  9. Applied add-cube-cbrt8.7

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

  10. Applied sqrt-prod8.7

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

  12. Applied add-cube-cbrt8.7

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

  13. Applied add-cube-cbrt8.7

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

  14. Applied times-frac8.7

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

  15. Final simplification8.7

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

Reproduce

herbie shell --seed 2019050 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))