Average Error: 12.6 → 7.6
Time: 2.7m
Precision: 64
Internal Precision: 320
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \left(\frac{D \cdot M}{2 \cdot d} \cdot h\right) \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\ell}}\]

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 12.6

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

  2. Initial simplification12.2

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

  4. Applied associate-/l*11.0

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

  6. Applied associate-/r/11.0

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

  8. Applied associate-/r/7.6

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

  9. Applied associate-*l*7.6

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

  10. Final simplification7.6

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

Runtime

Time bar (total: 2.7m)Debug logProfile

herbie shell --seed 2018250 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))