Average Error: 13.3 → 8.8
Time: 52.5s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[w0 \cdot \sqrt{1 - \frac{\left(\left(\left(M \cdot D\right) \cdot \frac{1}{2 \cdot d}\right) \cdot h\right) \cdot \frac{M \cdot D}{2 \cdot d}}{\ell}}\]

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 13.3

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

  3. Applied associate-*r/10.3

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

  5. Applied unpow210.3

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

  6. Applied associate-*l*8.8

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

  8. Applied div-inv8.8

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

  9. Final simplification8.8

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

Runtime

Time bar (total: 52.5s)Debug logProfile

herbie shell --seed 2018348 +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))))))