Average Error: 14.1 → 8.3
Time: 3.0m
Precision: 64
Internal Precision: 128
\[w0 \cdot \sqrt{1 - {\left(\frac{M \cdot D}{2 \cdot d}\right)}^{2} \cdot \frac{h}{\ell}}\]
↓
\[\sqrt{1 - \frac{\frac{D \cdot M}{2 \cdot d} \cdot h}{\ell} \cdot \frac{D \cdot M}{2 \cdot d}} \cdot w0\]

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

Derivation

  1. Initial program 14.1

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

  2. Simplified14.1

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

  4. Applied associate-*r/10.6

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

  6. Applied associate-*l*9.0

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

  8. Applied *-un-lft-identity9.0

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

  9. Applied times-frac8.3

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

  10. Final simplification8.3

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

Reproduce

herbie shell --seed 2019090 
(FPCore (w0 M D h l d)
  :name "Henrywood and Agarwal, Equation (9a)"
  (* w0 (sqrt (- 1 (* (pow (/ (* M D) (* 2 d)) 2) (/ h l))))))