Average Error: 14.3 → 9.3
Time: 1.6m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{\frac{D}{d} \cdot \frac{M}{2}}{\ell} \cdot \left(\frac{D}{d} \cdot \frac{M}{2}\right)}{\frac{1}{h}}} \cdot w0\]

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 14.3

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

  2. Initial simplification14.0

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

  4. Applied div-inv14.0

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

  5. Applied associate-/r*10.9

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

  7. Applied div-inv10.9

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

  9. Applied pow110.9

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

  10. Applied pow110.9

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

  11. Applied pow110.9

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

  12. Applied pow-prod-down10.9

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

  13. Applied pow-prod-down10.9

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

  14. Simplified9.3

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

  15. Final simplification9.3

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

Runtime

Time bar (total: 1.6m)Debug logProfile

herbie shell --seed 2018219 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))