Average Error: 14.0 → 8.9
Time: 54.0s
Precision: 64
Internal Precision: 576
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\left(\frac{D \cdot M}{2 \cdot d} \cdot h\right) \cdot \frac{D \cdot M}{2 \cdot d}}{\ell}}\]

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 14.0

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

  2. Initial simplification13.5

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

  4. Applied associate-/r/10.5

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

  6. Applied associate-*l/10.4

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

  8. Applied associate-*l*8.9

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

  9. Final simplification8.9

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

Runtime

Time bar (total: 54.0s)Debug logProfile

herbie shell --seed 2018227 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))