Average Error: 13.6 → 8.9
Time: 1.0m
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\frac{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}{\frac{\ell}{\frac{D \cdot M}{2 \cdot d}}}}{\frac{1}{h}}}\]

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. Initial simplification13.2

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

  4. Applied div-inv13.2

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

  5. Applied associate-/r*10.2

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

  7. Applied associate-/l*8.9

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

  9. Applied div-inv8.9

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

  10. Final simplification8.9

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

Runtime

Time bar (total: 1.0m)Debug logProfile

herbie shell --seed 2018274 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))