Average Error: 13.6 → 8.3
Time: 3.0m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - {\left(\left(h \cdot \left(\frac{M}{2} \cdot \frac{D}{d}\right)\right) \cdot \frac{\frac{M}{2} \cdot \frac{D}{d}}{\ell}\right)}^{1}}\]

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 13.6

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

  3. Applied div-inv13.6

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

  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 unpow210.2

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

  7. Applied associate-*l*8.7

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

  9. Applied times-frac9.6

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

  11. Applied pow19.6

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

  12. Applied pow19.6

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

  13. Applied pow19.6

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

  14. Applied pow19.6

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

  15. Applied pow-prod-down9.6

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

  16. Applied pow-prod-down9.6

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

  17. Applied pow-prod-down9.6

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

  18. Applied simplify8.3

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

Runtime

Time bar (total: 3.0m)Debug logProfile

herbie shell --seed 2018195 +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))))))