Average Error: 13.2 → 8.6
Time: 57.4s
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - h \cdot \frac{\frac{D \cdot M}{2 \cdot d}}{\frac{\ell}{\left(D \cdot M\right) \cdot \frac{1}{2 \cdot d}}}} \cdot w0\]

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 13.2

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

  2. Initial simplification12.8

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

  4. Applied associate-/r/10.2

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

  6. Applied associate-/l*8.6

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

  8. Applied div-inv8.6

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

  9. Final simplification8.6

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

Runtime

Time bar (total: 57.4s)Debug logProfile

herbie shell --seed 2018278 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))