Average Error: 13.5 → 9.7
Time: 1.4m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{1}{\ell} \cdot \left(\left(\left(\sqrt[3]{\sqrt[3]{\frac{\frac{D}{2}}{\frac{d}{M}}} \cdot \sqrt[3]{\frac{\frac{D}{2}}{\frac{d}{M}}}} \cdot \sqrt[3]{\sqrt[3]{\frac{\frac{D}{2}}{\frac{d}{M}}}}\right) \cdot \frac{\frac{D}{2}}{\frac{d}{M}}\right) \cdot \left({\left(\sqrt[3]{\frac{M \cdot D}{2 \cdot d}}\right)}^{2} \cdot h\right)\right)} \cdot w0\]

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

Try it out

Your Program's Arguments

Results

Enter valid numbers for all inputs

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. Using strategy rm

  3. Applied div-inv13.5

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

  4. Applied associate-*r*10.2

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

  6. Applied add-cube-cbrt10.2

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

  7. Applied unpow-prod-down10.2

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

  8. Applied associate-*l*8.8

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

  9. Simplified9.7

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

  11. Applied add-cube-cbrt9.7

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

  12. Applied cbrt-prod9.7

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

  13. Final simplification9.7

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

Runtime

Time bar (total: 1.4m)Debug logProfile

herbie shell --seed 2018216 +o rules:numerics
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))